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https://mathoverflow.net/questions/73385 | 25 | I'm teaching an introductory graph theory course in the Fall, which I'm excited about because it gives me the chance to improve my understanding of graphs (my work is in topology). A highlight for me will be to teach the Matrix-Tree Theorem, which I think is the only place that linear algebra is used in the course.
L... | https://mathoverflow.net/users/2051 | The Matrix-Tree Theorem without the matrix | A combinatorial proof of the matrix-tree theorem can be found in the paper by D. Zeilberger
[A combinatorial approach to matrix algebra,](http://www.math.rutgers.edu/~zeilberg/mamarimY/Zeilberger_y1985_p61.pdf)
Discrete Math. 56 (1985), 61–72.
The proof uses only the interpretation of the determinant as an alternatin... | 28 | https://mathoverflow.net/users/10744 | 73429 | 44,698 |
https://mathoverflow.net/questions/73431 | 4 | **EDIT**
Oops---I found the answer to the first question of mine [here on Wikipedia](http://en.wikipedia.org/wiki/Separable_space#Embedding_separable_metric_spaces)---this is really classic material. I'll leave the question open for a bit, in case someone tells me something interesting for my second question.
(A ve... | https://mathoverflow.net/users/8430 | When is a metric space isometrically embeddable into some Banach space? | Arbitrarily fix $y\_0\in X$. Then, with every $y\in X$ you can associate a bounded continuous function from X to R defined by
$$f\_y(x)=d(y,x)-d(y\_0,x).$$
It is easy to show that
$$\max\_x |f\_y(x)-f\_z(x)|=d(y,z),$$
with the maximum assumed if $x=y$ or $x=z$. Hence $X$ is isometrically embedded in the Banach space
$C... | 3 | https://mathoverflow.net/users/12120 | 73433 | 44,699 |
https://mathoverflow.net/questions/73439 | 14 | Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $A$, given a (smooth) matrix of 2-forms $F$ which satisfies the condition $dF =B \wedge F - F \wedge B$ for some smooth m... | https://mathoverflow.net/users/3709 | When is a given matrix of two forms a curvature form? | The answer is generally 'no'; for most $F$ that satisfy your condition, there will not exist an $A$ that satisfies $F = dA + A\wedge A$.
The easiest counterexample I know of is when $n=4$ and the matrix $F$ is $2$-by-$2$. To begin, note that you can reduce to the case when both $F$ and the $A$ you seek have trace ze... | 25 | https://mathoverflow.net/users/13972 | 73451 | 44,706 |
https://mathoverflow.net/questions/73454 | 6 | Can anyone help me with a proof of the following claim (see for example the book Higher algebraic geometry of Olivier Debarre, proof of Proposition 1.43, page 31):
Let X be a complex manifold, and let W be a complex submanifold of X, with codimension $\geq 2$. Let $\pi :Y \rightarrow X$ be a bimeromorphic morphism, w... | https://mathoverflow.net/users/17326 | Universal property of blowups | I am kind of a rookie at this, but what if Y is a small resolution of a double point on a threefold X, with one dimensional excepTional locus. Then it seems false to expect a factorization through the blowup since the curve exceptional locus could not map onto the two dimensional exceptional locus of the blowup of X. w... | 6 | https://mathoverflow.net/users/9449 | 73465 | 44,711 |
https://mathoverflow.net/questions/73447 | 9 | In the category of groups, it is elementary that all central extensions of a cyclic group are abelian. Is the same true, in the category of (finite?) group schemes over a field $k$, for central extensions of the group $\mu\_n$ of $n$th roots of unity?
| https://mathoverflow.net/users/6522 | Central extensions of group schemes | If we have a central extension of group schemes $1\rightarrow B \rightarrow C\rightarrow
A\rightarrow1$ with $A$ abelian, then we get a commutator mapping
$\Lambda^2A\rightarrow B$ (of sheaves as $\Lambda^2A$ in general is not a group
scheme) and the extension is abelian precisely when this map is zero. Hence for
an no... | 14 | https://mathoverflow.net/users/4008 | 73470 | 44,714 |
https://mathoverflow.net/questions/73419 | 1 | I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:
1. $K\_X$ is nef.
2. Every birational morphism from $X$ to $Y$ must be an isomorphism, where $Y$ is another variety with only terminal singularities.
Suppose $X$ is a... | https://mathoverflow.net/users/17314 | Why are the different definitions of minimal model equivalent? | 1 implies 2 follows from the so called "negativity lemma", see for example Lemma 3.39 in the book by Kollar-Mori. The main point is that an effective linear combination of exceptional divisors can never be $f$-nef for a birational morphism $f:X \to Y$.
2 implies 1 is not true: consider $X = \mathbb{P}^n$.
(The cond... | 5 | https://mathoverflow.net/users/519 | 73472 | 44,715 |
https://mathoverflow.net/questions/73450 | 15 | Let $X$ be a (quasi-)projective, nonsingular, complex variety. Denote by $\mathcal{T}\_X$ its tangent sheaf and by $X^{\mathrm{an}}$ its analytification. I am looking for a proof for the equality
$\displaystyle \int\_X c\_n(\mathcal{T}\_X) = \chi(X^{\mathrm{an}})$,
i.e. the degree of the top chern class is ... | https://mathoverflow.net/users/9947 | Top Chern Class = Euler Characteristic | As an alternative to R. Budney's answer, one might also notice that the Gauss-Bonnet formula (the one you mention - mind that you must assume that $X$ is projective, otherwise the integral might not even make sense) is a consequence of the Hirzebruch-Riemann-Roch theorem. Indeed, the HRR theorem says
$$
\chi(V)=\int\_{... | 23 | https://mathoverflow.net/users/17308 | 73474 | 44,716 |
https://mathoverflow.net/questions/73492 | 100 | I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
There is one strong argument against this: We tell students that $dy$ and $dx$ mean "a really small change... | https://mathoverflow.net/users/1050 | How misleading is it to regard $\frac{dy}{dx}$ as a fraction? | You can think of $x$ and $y$ as smooth functions on a one-dimensional manifold of states of some system that you are thinking about, then $dx$ and $dy$ are differential forms. In any open region where $dx$ does not vanish we can say that $dy/dx$ is the unique smooth function such that $(dy/dx)dx=dy$; in other words, $d... | 60 | https://mathoverflow.net/users/10366 | 73496 | 44,727 |
https://mathoverflow.net/questions/73494 | 2 | The problem of Schauder decomposition of a given Banach space seems to play an important role in the geometry of Banach spaces, especially when one is interested in finite dimensional Schauder decompositions (FDD).
I am wondering if the Schauder decomposition can be regarded (in special cases) as the internal counte... | https://mathoverflow.net/users/17338 | Recovering Schauder decompositions | No. You need the projections $Q\_n$ onto $E\_1\oplus \dots E\_n$ from $F$ to be uniformly bounded in order for $(E\_n)$ to be a Schauder decomposition for $F$. Even then $F$ need not be isomorphic to $c\_0$/$\ell\_p$. However, if the $Q\_n$ are uniformly bounded from $\ell\_p$, then by taking limits in the weak operato... | 2 | https://mathoverflow.net/users/2554 | 73507 | 44,737 |
https://mathoverflow.net/questions/73529 | 7 | I am interested in computing the (anti)-canonical class of the (total space of the) projective completion of the tautological bundle over $P^1\times P^1$. That is, the canonical class of $\mathbb P\_{P^1\times P^1}(J \oplus \mathscr O)$, where $J$ is the tautological line bundle on $P^1\times P^1$.
I believe this ca... | https://mathoverflow.net/users/17350 | How can one compute the canonical class of the projective completion of the tautological bundle over $P^1\times P^1$? | Why not use the Leray-Hirsch theorem? That says that the integral cohomology ring of a projectivized rank $n$ vector bundle $\pi: PE \to B$ is generated, as an algebra over the cohomology of the base $B$, by the first Chern class $h$ of the relative $O(1)$, with relation $h^n + c\_1 h^{n-1} + \dots + c\_n$, where $c\_i... | 19 | https://mathoverflow.net/users/6522 | 73531 | 44,749 |
https://mathoverflow.net/questions/73509 | 7 | I am trying to understand the construction of the Jacobian of a curve following the [notes of J. S. Milne](http://www.jmilne.org/math/CourseNotes/av.html)
The question is going to be about a particular step in the proof of Proposition 4.2b in Chapter III, but I will first briefly recall the setup.
Let $X$ be a sch... | https://mathoverflow.net/users/2234 | construction of the Jacobian of a curve | Let $N = L \otimes (q^\ast q\_\ast (L\otimes L\_\gamma^{-1}))^{-1}$. It suffices to show that the zero locus $D \subset C \times T$ of $s \in \Gamma(N)$ is flat over $T$. If $T$ is nice (Noetherian, blah, blah), it then suffices to show that the fiberwise degree of $D$ is constant. Note that the restriction of $N$ to $... | 2 | https://mathoverflow.net/users/83 | 73534 | 44,750 |
https://mathoverflow.net/questions/73526 | 50 | This is a soft question. How do people usually use arxiv to put their papers? At which stage does one usually put his/her paper/report there? Someone suggests me to submit a paper while putting it on arxiv. Is that the convention that people follow?
Thank you!
Anand
| https://mathoverflow.net/users/36814 | how to use arxiv? | My comments above formulated as an answer:
People typically post a preprint on the arxiv at the same time that they post it on their own homepage, with the goal of disseminating their work to their colleagues. (These days, posting on the web is more important than journal publication for sharing your work, and the ar... | 38 | https://mathoverflow.net/users/2874 | 73540 | 44,754 |
https://mathoverflow.net/questions/73312 | 1 | Let $M$ be a smooth manifold and $G\_k(M)$ be the $k$-dimensional Grassmian bundle of $M$. Let $K\subset M$ be a compact subset and $E:K\to G\_k(M)$ be a continuous distribution on $K$.
We say $E$ is integrable on $K$ if there exists a foliation $\mathcal{F}$ (or lamination, since it may only foliates a subset of $M$... | https://mathoverflow.net/users/11028 | Extension of integrable distribution over a subset | The answer in general is no.
If $K$ is a submanifold of $M$ then tangent bundle of $K$ defines an integrable distribution on $K$. To wit we are talking about the foliation with just one leaf: $K$.
If we can extend this foliation to a neighborhood of $K$ then the restriction of Bott's connection to $K$ induces a fl... | 2 | https://mathoverflow.net/users/605 | 73541 | 44,755 |
https://mathoverflow.net/questions/73488 | 4 | In many problems of enumerative combinatorics, one finds the solution formula that involve complex roots of unity, $\cos(\frac{n \pi }{ k})$ and $\sin(\frac{n \pi }{k})$. Can someone highlight any combinatorial interpretation of such expressions. I haven't find any book or paper highlighting this except some rudiments ... | https://mathoverflow.net/users/17336 | Combinatorial Interpretation | The appearance of roots of unity or $\cos(\frac{n\\pi}{k})$ and $\sin(\frac{n\pi}{k})$ in combinatorial contexts can almost always be explained through the representation theory of $\mathbb{Z}/n\mathbb Z$. The language of representation theory is avoided most of the time, and one attributes the appearance of $\cos(\fra... | 12 | https://mathoverflow.net/users/2384 | 73543 | 44,757 |
https://mathoverflow.net/questions/73545 | 3 | Hello, can somebody help with the following question that I have thought over for quite some time, to no avail?
Suppose f: X--->Y is a universal cover and g: Y--->Z a fiber bundle, where X, Y and Z are manifolds. Is the composition gof: X--->Z necessarily a fiber bundle?
THanks!
| https://mathoverflow.net/users/17356 | composition of covering map and bundle projection | It is better to just assume that $f$ is a covering space. By shrinking $Z$ we may assume that that $Z$ is a ball and that $Y=Z\times F$. As $X\to Y$ is a covering space and $Z$ is simply-connected there is a covering space $X'\to F$ such that $X\to Y$ is isomorphic to $Z\times X'\to Z\times F$ which gives what you want... | 5 | https://mathoverflow.net/users/4008 | 73546 | 44,759 |
https://mathoverflow.net/questions/73511 | 2 | **Update: problem reformulation**
---------------------------------
Following the advice in comments, I now restate my problem using Voronoi
tessellation.
Given a unit hypercube $H\_n=\{(x\_1,\ldots,x\_n)\in \mathbb{R}^n: 0\leq x\_i\leq
1\}$, generate $K$ random points in $H\_n$ using
uniform Poisson point process ... | https://mathoverflow.net/users/17332 | Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? | I'm not saying the distribution of $L$ is inappropriate, but I think it will be more easy to work with another one.
Let me give an answer that works with a general class of distributions. I also only assume that the tessellation is only made from convex polytopes.
First remark that if I denote by $T$ the union of a... | 2 | https://mathoverflow.net/users/16934 | 73561 | 44,765 |
https://mathoverflow.net/questions/73557 | 5 | Let $X$ be a variety defined over a number field $k$. If I blow-up along some arbitrary subvariety of $X$, what are the possible outcomes for the dimension of the singular locus of the variety? If the subvariety lies outside the singular locus of $X$, then it stays the same, if it is carefully chosen, it might go down.... | https://mathoverflow.net/users/17363 | Blowing up a subvariety - what can happen to the singular locus? | Any birational map $\pi:X'\to X$ is the blow-up of some ideal sheaf on $X$, so in general one must expect singularities on $X'$, even if the ideal is reduced (as you assume).
As a concrete example, let $X=\mathbb{A}^n$ and blow-up the complete intersection subvariety given by the ideal $I=(f,g)\subset k[x\_1,\ldots,... | 4 | https://mathoverflow.net/users/3996 | 73569 | 44,769 |
https://mathoverflow.net/questions/73550 | 3 | I have a reference request concerning Proposition 1.6 in the following article [Link](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-6/issue-3/Ergodic-theory-group-representations-and-rigidity/bams/1183548783.full)
The setting: Let $G$ be a locally compact, second c... | https://mathoverflow.net/users/10400 | Ergodic decomposition of quasi-invariant measure | This is a bit too long for a comment, hence I post it as an answer.
I honestly don't know where you can find a group theoretic version of ergodic decomposition proved via Choquet theory (and I'm not convinced that it exists in the setting you're interested in).
However, the exact result you quote from Zimmer is pro... | 6 | https://mathoverflow.net/users/11081 | 73576 | 44,775 |
https://mathoverflow.net/questions/73571 | 4 | Dear All!
I tried for several evenings to find an answer to the following basic question and I cannot see what is the answer:
Given an integer $n\geq 3$, does there exist an (infinite) group with exactly $n$ normal subgroups?
If "yes", what about the same questions for finitely generated groups, finitely presente... | https://mathoverflow.net/users/13070 | (F.g., f.p.) groups with exactly $n$ normal subgroups | If $n$ is even, then the answer is "yes". Take the direct product of a simple (infinite) group and ${\mathbb Z}/2^j{\mathbb Z}$. Every normal subgroup either is inside the finite cyclic group or contains the simple group. Total number is twice the number of normal subgroups of the cyclic group.If $n$ is odd, you would ... | 8 | https://mathoverflow.net/users/nan | 73580 | 44,776 |
https://mathoverflow.net/questions/73566 | 6 | Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable Hilbert space which is closed under pointwise multiplication. The operator I'm actually looking at is a symmetric Marko... | https://mathoverflow.net/users/12366 | Literature on behaviour of eigenfunctions under multiplication? | It can easily happen that the product of two eigenfunctions has an infinite eigenfunction expansion. Probably this is more *typical* (in natural problems) than not. For example, the Laplace-Beltrami operator on compact Riemannian manifolds (or suitable values $(\Delta+c)^{-1}$ of its resolvent if one must have a *bound... | 7 | https://mathoverflow.net/users/15629 | 73581 | 44,777 |
https://mathoverflow.net/questions/73568 | 10 | Dear All!
At the time when Lyndon and Schupp wrote their book there was an open question:
Question: Does every finitely presented group with soluble word problem embed in a finitely presented simple group?
Is it still open? Could you hint at some useful references about this? Thanks!
| https://mathoverflow.net/users/13070 | Embedding in f.p. simple groups | I believe it is still open. By the Boone-Higman Theorem (W. W. Boone and G. Higman, "An algebraic characterization of the solvability of the word problem", J. Austral. Math. Soc. 18, 41-53 (1974)), a finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embed... | 9 | https://mathoverflow.net/users/35840 | 73585 | 44,780 |
https://mathoverflow.net/questions/73411 | 3 | Can one determine whether a given Eisenstein series ( for GL\_{2}(Q)) is overconvergent, just by looking at the associated Galois representation?
| https://mathoverflow.net/users/16673 | Eisenstein series and overconvergence | In the paper "Lissite de la Courbe de Hecke aux points Eisenstein critiques", Bellaiche and Chenevier completely classify all reducible Galois representations coming from overconvergent eigenforms of tame level 1. They are precisely those coming from either the ordinary family of Eisenstein series, or one of the "criti... | 4 | https://mathoverflow.net/users/5513 | 73592 | 44,784 |
https://mathoverflow.net/questions/72829 | 11 | Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h\_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$.
**Question 1.** Can one classify or describe the curves $Y$ such that $h\_{\textrm{Fal}}(Y) \geq 1$?
**Question 2.** For any $g>0$, does there exist a curve $Y$ of g... | https://mathoverflow.net/users/4333 | Which curves have stable Faltings height greater or equal to 1 | Dear Ariyan, the elliptic curve with equation $$y^2=x^3+6$$ has Faltings height
$$-(3/2)\log(\Gamma(1/3)/\Gamma(2/3))+(1/4)\log(3)=-0.748752...;$$ the curve
of genus $2$ with equation $$y^2+y=x^5$$ has Faltings height
$$
h\_{\rm Fal}(C\_{\bar{\bf Q}})=2\log(2\pi)-
{1\over 2}\log\big(\Gamma(1/5)^5\Gamma(2/5)^3\Gamma(... | 15 | https://mathoverflow.net/users/17308 | 73594 | 44,785 |
https://mathoverflow.net/questions/73596 | 4 | Sorry if this is too simple. This is my first question here.
Suppose $f : R^n \to R$ is a differentiable function. Say that we can compute in $T$ arithmetic operations the value $f(x)$ at any point $x$. Can we use that to somehow precisely bound the time that is required to compute $\nabla f$? (Intuitively because of... | https://mathoverflow.net/users/17366 | Complexity of computing derivatives | The complexity is $O(nT)$. Look up "automatic differentiation" in Wikipedia. This is taking your statement about computing $f(x)$ in $T$ arithmetic operations literally: the "arithmetic operations" could include arbitrary powers and elementary functions such as exp, ln, sin, considered as single operations, as long as ... | 12 | https://mathoverflow.net/users/13650 | 73602 | 44,789 |
https://mathoverflow.net/questions/73601 | 1 | How can you decide, if a polynomial with integer coefficients p(x) is the product (q(x))^2\*r(x) of two other polynomial with integer coefficients q(x), r(x)?
| https://mathoverflow.net/users/17367 | Polyomial roots | First of all, there is always the trivial factorization with $q=1$ and $r=p$. I'll discuss when more interesting ones exist, and how to find them. To make life simple, I'll deal with rational coefficients for most of the answer, and only address integrality at the end.
To give a short answer first, there is a nontri... | 7 | https://mathoverflow.net/users/297 | 73605 | 44,791 |
https://mathoverflow.net/questions/73533 | 3 | Checking a recent article [[this one](http://arxiv.org/abs/1108.1552), specifically section 3.1] I found the following claim (I'm paraphrasing, of course):
>
> Let $A$ be a graded connected
> noetherian algebra (not necessarily
> commutative), and suppose it is
> AS-Cohen-Macaulay of depth $d$. If $M$
> is a f... | https://mathoverflow.net/users/17353 | Local Cohomology and Maximal-Cohen-Macaulay modules | Well, I don't know if I'm supposed to, but since I found a solution, I'll write the general idea here.
[This is from an unpublished manuscript by P. Smith, the first author of the paper]: If $A$ is CM, let $\omega\_A = H^d\_\mathfrak m(A)^\*$ be its dualizing module. Then there is a spectral sequence
$$ E^{pq}\_2 = ... | 2 | https://mathoverflow.net/users/17353 | 73612 | 44,795 |
https://mathoverflow.net/questions/73613 | 6 | It is relatively easy to show that
$$
\sum\_{a\_1 + \cdots + a\_k=\ell} \binom{\ell}{a\_1,\ldots,a\_k} = k^\ell
$$
where $\binom{\ell}{a\_1, \ldots, a\_k} = \frac{\ell!}{a\_1!\cdots a\_k!}$. What can be said if we want to compute the restricted sum
$$
s(\ell,k) = \sum\_{a\_1 + \cdots + a\_k=\ell} \binom{\ell}{a\_1,\ldo... | https://mathoverflow.net/users/1703 | What is this restricted sum of multinomial coefficients? | $\binom{\ell}{a\_1,\dots,a\_k}$ is the coefficient of $x\_1^{a\_1}\cdots x\_k^{a\_k}$ in the expansion of
$$(x\_1 + x\_2 + \dots + x\_k)^{\ell}.$$
The sum of all these coefficients is obtained by substituting $x\_1=\dots=x\_k=1$.
To eliminate even $a\_1$, we can consider the expansion of
$$\frac{1}{2}(x\_1 + x\_2 + ... | 13 | https://mathoverflow.net/users/7076 | 73616 | 44,797 |
https://mathoverflow.net/questions/71657 | 5 | Does anyone know of a good reference describing the action of the Steenrod algebra $\mathcal{A}\_2$ on the cohomology algebra $$H^\ast(BO(k);\mathbb{F}\_2)\cong\mathbb{F}\_2[w\_1,w\_2,\ldots ,w\_k]$$ of the classifiying space for $k$-dimensional vector bundles? This is a polynomial algebra on the universal Stiefel-Whit... | https://mathoverflow.net/users/8103 | Steenrod squares in the cohomology of $BO(k)$ | In the linked paper, David and I have a self-contained, elementary proof of Theorem 2.3, i.e. the freeness of the Steenrod action, from which follow Remarks 2.4 and 2.5. (It does not use Lannes-Zarati, we mentioned their result only for completeness.)
| 2 | https://mathoverflow.net/users/17371 | 73620 | 44,800 |
https://mathoverflow.net/questions/73621 | 7 | Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This sequence appears to start 0, 1, 3, 5, 7, 9, 12, 14, 16...
What is the maximum number of triangular lattice cells? (Not th... | https://mathoverflow.net/users/9021 | Maximal number of edges and triangular cells for n points in a triangular lattice | The following was conjectured by D. Reutter in problem 664A, Elemente der mathematik 27 and proved by H. Harborth in Solution to problem 664A, Elemente der mathematik 29, 14-15
>
> The maximum number of times the minimum distance can occur among $n$ points in the plane is $\lfloor 3n-\sqrt{12n-3}\rfloor$.
>
>
>
... | 10 | https://mathoverflow.net/users/2384 | 73625 | 44,803 |
https://mathoverflow.net/questions/73603 | 3 | How can one calculate the index of a Fredholm operator numerically ?
In numerically calculations one uses always finte dimensional spaces.
But linear operators on finite dimensional spaces have always index zero.
| https://mathoverflow.net/users/17261 | How to calculate a Fredholm index numerically | The two key properties of the Fredholm index are
* It is a (norm)-continuous function from the bounded linear operators to the integers. In particular, if $A$ is a Fredholm operator, then there exists $\delta > 0$ such that for $\|A - B\| < \delta$, we have $index(A) = index(B)$. This tells you that you can approxima... | 2 | https://mathoverflow.net/users/3983 | 73635 | 44,811 |
https://mathoverflow.net/questions/73599 | 3 | I found it very hard to find literature about smooth manifolds that are not required to be Hausdorff. In particular I'm interested in their local properties:
1.) Are the $r$-th order jet bundles $J^r(M,N)$ well defined for non Hausdorff manifolds?
(Recall that this question includes the tangent bundle as it is $J^1(\... | https://mathoverflow.net/users/17267 | Jet spaces between non Hausdorff manifolds | It doesn't contain any proofs, but Bourbaki's *Variétés différentiables - Fascicule de résultats* defines jet bundles (Section 12) without assuming that the underlying varieties are Hausdorff.
| 2 | https://mathoverflow.net/users/10696 | 73641 | 44,814 |
https://mathoverflow.net/questions/73634 | 4 | One knows that the support $S$ of a coherent sheaf on a noetherian scheme is closed.
E.g. on an affine scheme $X=Spec(A)$ and $F$ corresponding to a finitely generated $A$-module $M$, then the closed subset which corresponds to $S$ is just $V(Ann(M))$.
One often says that $S$ is endowed with the structure of a closed... | https://mathoverflow.net/users/16876 | subscheme structure of support | No it isn't the reduced induced closed subscheme structure in general.
For example, let $A={\bf Z}$, $M={\bf Z}/4{\bf Z}$. Then ${\rm Ann}(M)=4{\bf Z}$ and the prime ideal defining $S$ (with its reduced structure) is $2{\bf Z}={\rm rad}({\rm Ann}(M))$. So if $S$ is endowed with the reduced structure, it is isomorphic ... | 4 | https://mathoverflow.net/users/17308 | 73644 | 44,817 |
https://mathoverflow.net/questions/73652 | 5 | Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\operatorname{char}(k) = 0$, equipped with:
* a commutative associative $k$-linear multiplication $\circ\\,$;
* a positive-d... | https://mathoverflow.net/users/12858 | adjoint of multiplication operator in a commutative algebra | In case $k=\mathbb{C}$, what you're describing is a finite-dimensional *H\*-algebra*. More generally, these are Banach algebras, whose carrier space is a Hilbert space, satisfying the adjoint property you mention.
It is natural to make a nondegeneracy assumption: $A$ is called *proper* when $\forall a \in A\\,.\\, a... | 9 | https://mathoverflow.net/users/10368 | 73655 | 44,820 |
https://mathoverflow.net/questions/73649 | 12 | This question has been asked on MathExchange to no avail.
Suppose $G$ is a finitely generated nilpotent group with abelianization of rank $r$. Does $G$ always have a subgroup $H$ of finite index, such that $H$ abelianized is a free abelian group of rank $r$?
Since this is MathOverflow, I will push the question furt... | https://mathoverflow.net/users/17378 | Finite index subgroup with free abelianization | **Warning:** (YCor) the following argument is mistaken as was pointed out by Derek Holt: the assertion that the abelianization of a torsion-free nilpotent group is torsion-free is hopelessly wrong.
---
The answer is "yes" because every f.g. nilpotent group has a torsion-free finite index subgroup and because the ... | -1 | https://mathoverflow.net/users/nan | 73663 | 44,825 |
https://mathoverflow.net/questions/73664 | 16 | We all know [what polynomials are](http://en.wikipedia.org/wiki/Polynomial), along with their elementary properties and many effective algorithms for different representations of polynomials.
The question here is more of a *universal algebra* question: what is the signature of the theory which best corresponds to pol... | https://mathoverflow.net/users/3993 | What is the theory of polynomials? | My first reaction to this is that you might be interested in the general theory of [Tall-Wraith monoids](http://ncatlab.org/nlab/show/Tall-Wraith+monoid). The special case which might be of more immediate interest to you is the notion of **plethory**. A good reference is [this paper](http://maths.anu.edu.au/~borger/pap... | 11 | https://mathoverflow.net/users/2926 | 73665 | 44,826 |
https://mathoverflow.net/questions/73659 | 9 | One of the motivations to study tropical geometry is that there are some hard Algebraic Questions that can be answered by proving them in the Tropical World. For example one can show that tropical Bezout's Theorem implies the Algebraic Bezout.
What properties are there known that are true (or might be) in tropical g... | https://mathoverflow.net/users/13782 | Properties from Tropical Geometry that do not imply their algebraic counterpart. | There is a simple nice fact which holds in the tropical plane that has no counterpart in algebraic geometry (nor in any kind of standard geometry I might think of): two tropical lines always "intersect" in a single point... even if they coincide!
Of course this property relies on the fact that "intersection" is not d... | 7 | https://mathoverflow.net/users/6205 | 73667 | 44,827 |
https://mathoverflow.net/questions/73675 | 7 | Dear all,
I'm seeking a reference for a claim made in lecture 8 of Jacob Lurie's chromatic homotopy theory notes (<http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf>). More particularly, Theorem 6 of this lecture states that (say over $\mathbb{F}\_2$, so that things are commutative) the spectrum $\mathbb{G} =... | https://mathoverflow.net/users/1202 | Reference request: Spec A_* is the automorphism group of the additive formal group law | MIT OpenCourseWare has some [notes from a course](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/) that Lurie taught in 2007. I believe [the lecture on the dual Steenrod algebra](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic... | 4 | https://mathoverflow.net/users/121 | 73677 | 44,830 |
https://mathoverflow.net/questions/73673 | 5 | Suppose we have symplectic manifolds $(M\_1, \omega\_1)$ and $(M\_2, \omega\_2)$ with non-empty boundary of contact . Often we need to deal with the product $M\_1 \times M\_2$ with the product symplectic structure. Can we round the corners to get a contact manifold as boundary?
| https://mathoverflow.net/users/5538 | "Rounding the corners" to get contact boundary | In that generality, the answer is no: a symplectic form $\omega$ on $X$ which has contact-type boundary is exact on $\partial X$. Yet $\omega\_1 \oplus \omega\_2$ need not be exact on $M\_1\times \partial M\_2$, nor on $\partial M\_1 \times M\_2$.
It is possible, however, if $M\_1$ and $M\_2$ are Liouville domains, i... | 6 | https://mathoverflow.net/users/2356 | 73678 | 44,831 |
https://mathoverflow.net/questions/73674 | 6 | I had a hard time trying to solve exercise 7.24 in Jech's book (3rd edition, 2003) and finally came to the conclusion that the result there, which should be proved might be wrong. The claim goes like this:
Let $A$ be a subalgebra of a Boolean algebra $B$ and suppose that $u \in B-A$. Then there exist ultrafilters $F,... | https://mathoverflow.net/users/4753 | An exercise in Jech's Set Theory | Your counterexample is not correct. Let $r$ be an irrational real number, and let $F$ be the principal ultrafilter in $B$ on the closed interval $[r,r]=\{r\}$, which is an atom in $B$. Note that $F\cap A$ is the ultrafilter of all elements of $A$ in which $r$ is a member. Now consider the complement $-[r,r]=(-\infty,r)... | 11 | https://mathoverflow.net/users/1946 | 73679 | 44,832 |
https://mathoverflow.net/questions/73484 | 1 | I have a couple questions regarding the proof of Proposition $3$ (see page $10-11$ of arxiv.org/abs/math/0102039) in Bezrukavnikov's paper "Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone" . For simplicity, assume $G$ is simply connected, simple algebraic group (so that $G$ is its own univ... | https://mathoverflow.net/users/2623 | Pushforwards/pullbacks of some line bundles on (partial) flag varieties | For question 1, let $L\_\alpha$ denote the Levi of the parabolic $P\_\alpha$; then $L\_\alpha$ has derived group $SL\_2$. Let $B\_\alpha$ denote the Borel of $L\_\alpha$ such that $B \cap L\_\alpha = B\_\alpha$. Note that $V\_{\lambda'} = p\_\alpha^\* O\_{G/B}(V\_\alpha(\lambda'))$, where $V\_\alpha(\lambda')$ is the $... | 2 | https://mathoverflow.net/users/1528 | 73680 | 44,833 |
https://mathoverflow.net/questions/73676 | 2 | Consider the linear (geometric) wave equation in dimension (3+1) with non smooth background metric $g$ say $g \in L^\infty\_t H^3\_x$ and $\partial\_t g \in L^\infty H^2\_x$, then energy estimates enable to propagate sobolev regularity of the initial data set till $H^3\times H^2$, is this sharp in the following sense :... | https://mathoverflow.net/users/17388 | optimality of energy estimates for non smooth metric | -> question reedited
| 0 | https://mathoverflow.net/users/17384 | 73683 | 44,835 |
https://mathoverflow.net/questions/73483 | 9 | We're looking for a large set of exact sequences of vector bundles on Grassmannians. Here's the set up:
$V$ and $Q$ are complex vector spaces of dimensions $d$ and $r$ respectively $(d\geq r)$, and we're working on the Grassmannian $Gr(V,Q)$. For simplicity let's fix a trivialization of $det(V)$.
Now let $\alpha$ b... | https://mathoverflow.net/users/2454 | Exact sequences of bundles on Grassmannians | Look at Fonarev's [*On minimal Lefschetz decompositions for Grassmannians*](https://arxiv.org/abs/1108.2292v1), specifically Proposition 5.3 ([link to proposition in the PDF](https://arxiv.org/pdf/1108.2292v1.pdf#page=14)). I guess this exact sequence is what you need.
| 3 | https://mathoverflow.net/users/4428 | 73690 | 44,838 |
https://mathoverflow.net/questions/73681 | 11 | This is a simple bibliographic request that I have been unable to pin down. Max Dehn's
solution to Hilbert's 3rd problem is:
>
> Max Dehn, "Über den Rauminhalt." *Mathematische Annalen* **55** (190x), no. 3, pages 465–478.
>
>
>
It is variously cited as either 1901 or 1902 (but always volume 55; Hilbert's own ... | https://mathoverflow.net/users/6094 | Dehn's solution to Hilbert's 3rd: 1901 or 1902? | the journal has been scanned and can be read here:
<https://archive.org/details/mathematischean33behngoog>
volume 55 has four issues, covering both years 1901 and 1902; that is where the confusion comes from; Dehn's article is from the third issue, published in September 1901.
you can read the table of contents h... | 16 | https://mathoverflow.net/users/11260 | 73695 | 44,842 |
https://mathoverflow.net/questions/73687 | 10 | It is quite obvious that if a map is a homotopy equivalence, then its mapping cone is contractible, but is the converse true: mapping cone contractible => the map is a homotopy equivalence? I am thinking about both the topological category and the category of chain complexes.
| https://mathoverflow.net/users/9800 | when mapping cone is contractible | Ok, going off my second comment from above, in exercise 9 from section 4.2 Hatcher gives a hint that solves your problem. Let $X$ be an acyclic CW-complex which isn’t contractible (I'll give an example below to be complete). Let $f: X \rightarrow \*$. The mapping cone of this is $SX$, the suspension of $X$. Exercise 8 ... | 17 | https://mathoverflow.net/users/11540 | 73702 | 44,846 |
https://mathoverflow.net/questions/73669 | 1 | Hi, this is related to [this](https://mathoverflow.net/questions/59661/probability-of-return-at-step-n-of-a-random-walk-to-its-starting-vertex) earlier question.
Given Random walk on a regular graph $G=(V,E)$. The Random walk is simple so that transition probabilities are $1/\text{deg}(v\_i)$, and time is in discrete... | https://mathoverflow.net/users/13932 | Probability of first return to starting vertex in Random walk on regular finite graph | For a regular graph, each walk of a given length has the same probability, so let's just consider the number of walks.
A walk starting and ending at a given vertex is comprised of zero or more pieces that consist of non-trivial walks that return to the start only on their last step. So if $w(x)$ is the ordinary gener... | 3 | https://mathoverflow.net/users/9025 | 73709 | 44,849 |
https://mathoverflow.net/questions/73719 | 4 | This is a topological question that came up tangentially to some material I was working on. Suppose $X$ and $Y$ are complete metric spaces and $D$ is a dense subset of $X$. Let $f:D\mapsto Y$ be a continuous injection. Extend $f$ to a function $g:X\mapsto Y$ by continuity. Must $g$ be injective? It seems to me that the... | https://mathoverflow.net/users/936 | Injective Function on a Dense Set | Map the open unit interval to a circle minus a point, and then extend it to the closed interval.
| 12 | https://mathoverflow.net/users/1946 | 73721 | 44,856 |
https://mathoverflow.net/questions/73716 | 4 | Let $\mathfrak{g}$ be a semisimple Lie algebra and $V\_{\lambda}$ be the irreducible $\mathfrak{g}$-module with highest weight $\lambda$. Are there some softwares which can compute the formal character $ch(V\_{\lambda})=\sum\_{\mu}dim(V\_{\mu})e(\mu)$ explicitly? We can use Weyl formula. But it is difficult to compute ... | https://mathoverflow.net/users/11877 | Compute formal character of semisimple Lie algebras. | The software package [LiE](http://www-math.univ-poitiers.fr/~maavl/LiE/) is good for this. It is no longer maintained, but it has a lot of functionality, and the documentation is good. There is an online demonstration [here](http://www-math.univ-poitiers.fr/~maavl/LiE/form.html). There is a bit of a learning curve, but... | 7 | https://mathoverflow.net/users/703 | 73723 | 44,858 |
https://mathoverflow.net/questions/73671 | 7 | Let $C$ be a curve of genus $g \geq 1$ and let $J^d$ be its degree $d$ Jacobian.
Inside of $J^{g-1}$ there is the Theta divisor $\Theta$, which can be defined in various ways; the quickest definition is probably: it's the image of the Abel-Jacobi map $C^{(g-1)} \to J^{g-1}$ sending an effective degree $g-1$ divisor t... | https://mathoverflow.net/users/83 | Cohomology of Theta divisor on Jacobian? | this seems to be the kodaira vanishing theorem. i.e. any line bundle of form K+A where is ample, has no higher cohomology. for an abelian variety K is trivial, and Theta is ample. qed.
| 5 | https://mathoverflow.net/users/9449 | 73727 | 44,860 |
https://mathoverflow.net/questions/19170 | 26 | In [Number of digits in n!](https://mathoverflow.net/questions/19086), now closed, there was a mention of Dmitry Kamenetsky's formula, $[\bigl(\log(2\pi n)/2+n(\log n-\log e)\bigr)/\log 10]+1$, for the number of decimal digits in $n$-factorial. Here, $[x]$ is the integer part of $x$. The formula appears at A034886 in t... | https://mathoverflow.net/users/3684 | How good is Kamenetsky's formula for the number of digits in n-factorial? | A counterexample is $n\_1 := 6561101970383$, with
$$
\log\_{10} \left( (n\_1/e)^{n\_1} \sqrt{2\pi n\_1} \right)
= 81244041273652.999999999999995102483 - \phantom; ,
$$
but
$$
\log\_{10} (n\_1!)
= 81244041273653.000000000000000618508 + \phantom;.
$$
If I computed correctly, $n\_1$ is the first counterexample, and the on... | 70 | https://mathoverflow.net/users/14830 | 73730 | 44,862 |
https://mathoverflow.net/questions/73726 | 3 | Goal:
I want to generate a r-regular graph with n vertices. rn = 2m.
Current best:
```
(1) take n vertices; randomly pick a vertex v of degree < r.
(2) S = set of all vertices of degree < r, and not a neighbor of v.
(3) create an edge between v and a random element of S.
(4) repeat.
```
Question:
Is there a ... | https://mathoverflow.net/users/17394 | Generating r-Regular Random Graph in Parallel | This question is more difficult that it seems.
Firstly, there is a difference between picking edges of a graph uniformly, and picking a $r$-regular graph uniformly.
Let $G\_{r,n}$ be the set of $r$-regular graphs on $n$ nodes. By "uniformly pick a $r$-regular graph", you need to create an algorithm that chooses $G ... | 4 | https://mathoverflow.net/users/8769 | 73735 | 44,864 |
https://mathoverflow.net/questions/73741 | 6 | One knows that in higher category theory, the category of $(\infty,n-1)$ categories is naturally an $(\infty,n)$ category ,(I use the word category to mean category in the correct weakened sense). When the category of $(\infty,1)$ categories is regarded as a weakened kan complex, we may regard these objects as a full s... | https://mathoverflow.net/users/16801 | What does the "category" of $(\infty,1)$ category look like. | You can see the collection of $(\infty,1)$-categories as forming themselves an $(\infty,1)$-category, which is sufficient to see where weak associativity shows up: There are many models for the intuiti9ve concept of $(\infty,1)$-category, the simplest is that of a usual 1-category endowed with a class of weak equivalen... | 11 | https://mathoverflow.net/users/733 | 73755 | 44,874 |
https://mathoverflow.net/questions/73751 | 11 | Let $F$ be any field of zero characteristic, $F^{\ast}$ its multiplicative group and $T$ is the torsion group.
Is it true that $T$ is a direct summand for $F^{\ast}$?
| https://mathoverflow.net/users/17399 | torsion group of the multiplicative group of a field | This was a problem that was asked by Fuchs in his book "Abelian groups" (1958). It was first solved in negative by P. M. Cohn in "Eine Bemerkung uber die multiplikative Gruppe eines Korpers", Arch. Math. (Basel) 13 (1962) 344-48. ([MR0146252](http://www.ams.org/mathscinet-getitem?mr=146252)). Later W. May gave a counte... | 14 | https://mathoverflow.net/users/2384 | 73757 | 44,875 |
https://mathoverflow.net/questions/73651 | 43 | The graph [reconstruction conjecture](https://en.wikipedia.org/wiki/Reconstruction_conjecture) claims that (barring trivial examples) a graph on n vertices is determined (up to isomorphism) by its collection of (n-1)-vertex induced subgraphs (again up to isomorphism).
The way it is phrased ("reconstruction") suggests... | https://mathoverflow.net/users/1492 | True by accident (and therefore not amenable to proof) | This is a very interesting (yet rather vague) question. Most answers were in the direction of mathematical logic but I am not sure this is the only (or even the most appropriate) way to think about it. The notion of coincidence is by itself very complicated. (See <https://en.wikipedia.org/wiki/Coincidence> ). One way t... | 24 | https://mathoverflow.net/users/1532 | 73768 | 44,882 |
https://mathoverflow.net/questions/73760 | 2 | Let $1\to H\to E\to G\to 1$ be a short exact sequence of algebraic groups defined over an algebraically closed field $k$ of characteristic $p$. Suppose $H$ is a finite group, and $G$ and $E$ are connected. Does it follow that $G\cong E$?
[Edit: of course not in general, since, as Max points out, E=SL\_n, H=Z(E) and G... | https://mathoverflow.net/users/16185 | Connected extensions of finite by connected algebraic groups | In Groupes algébriques et corps de classes Serre classifies the $2$-dimensional commutative unipotent connected algebraic groups $G$ (VII:11). With the exception of the product of the additive group with itself they are all isogenous to the Witt vector group $W\_2$ so that there is an exact sequence as per above with $... | 6 | https://mathoverflow.net/users/4008 | 73769 | 44,883 |
https://mathoverflow.net/questions/73717 | 2 | Mathematics is the universal language.
That is, until someone says the word "obvious", or "well known". At which point it becomes relative to the reader.
My question is about a "well known" theorem. My problem is that it is not known to me. But I would like to know.
The following comes from Y. Katznelson and B. ... | https://mathoverflow.net/users/8769 | Point mapping induces a set mapping | This reference seems to be exhaustive (and exhausting) on the subject:
<http://matwbn.icm.edu.pl/ksiazki/cm/cm2/cm2131.pdf>
| 1 | https://mathoverflow.net/users/11142 | 73777 | 44,889 |
https://mathoverflow.net/questions/73761 | -2 | From Alan Turing we know what we can expect from a computer and from Claude Shannon what we can expect from a communication channel.
Does anyone know any connection between these two theories (namely, Automata Theory and Information Theory) which actually set the theoretical limits of the nowadays information technol... | https://mathoverflow.net/users/11825 | Turing-Shannon connection | See
<http://en.wikipedia.org/wiki/Algorithmic_information_theory>
| 2 | https://mathoverflow.net/users/11142 | 73778 | 44,890 |
https://mathoverflow.net/questions/73780 | 4 | I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ that's not sheaf, the presheaf of constant real-valued functions on $\mathbb{R}^{2}$. Let the topological space $X = \mathbb{R}^{2}$ ... | https://mathoverflow.net/users/13707 | Are presheaves of constant functions sheaves? | As you say yourself, the overlap condition is vacuous and thus automatically true. However, the sheaf condition for presheaves has two parts:
* The overlap condition: that $\sigma\_i | (U\_i \cap U\_j) = \sigma\_j | (U\_i \cap U\_j)$. This is true, since the intersection is empty.
* The gluing condition: that there i... | 14 | https://mathoverflow.net/users/6545 | 73781 | 44,892 |
https://mathoverflow.net/questions/73792 | 7 | Assume $f\colon \mathbb Q\to \mathbb Q$ is a function which admits continuous extensions
* $f\_0\colon\mathbb R\to \mathbb R$ and
* $f\_p\colon \mathbb Q\_p\to \mathbb Q\_p$ for each prime $p$.
>
> Is it true that $f$ is a polynomial?
>
>
>
I guess the answer is **no**, but I do not see a counterexample.
| https://mathoverflow.net/users/10330 | Continuous extensions reals and to p-adic numbers | The answer is no, and one can essentially use the same construction as in the answer:
[Is a real power series that maps rationals to rationals defined by a rational function?](https://mathoverflow.net/questions/42460)
Specifically, enumerate the non-zero rationals $\{r\_1,r\_2, \ldots\}$ in some way. Now consider the... | 14 | https://mathoverflow.net/users/nan | 73803 | 44,909 |
https://mathoverflow.net/questions/73805 | 21 | The set of $n\times n$ real, nonnegative matrices whose rows and columns sum to one forms the well-known [*Birkhoff polytope*](http://en.wikipedia.org/wiki/Birkhoff_polytope)
Recently someone asked me if I knew
>
> How to sample (in polynomial time) uniformly at random, from the Birkhoff polytope?
>
>
>
Clea... | https://mathoverflow.net/users/8430 | Sampling from the Birkhoff polytope | This is, to my knowledge, still open. It is connected to the problem of computing the volume of the Birkhoff polytope (or computing the volume of its faces), which is known in closed form only for $n\le 14$. ~~This is also equivalent to~~This could be approached by counting non-negative integer matrices with equal row ... | 17 | https://mathoverflow.net/users/2384 | 73811 | 44,912 |
https://mathoverflow.net/questions/73818 | 15 | I know classification of 2 manifolds and geometrization for 3 manifolds.
Why for dimension great or equal to 4, this task become impossible?
edit: Or should I ask "why geometrization won't be possible for 4 or higher dimension?"
| https://mathoverflow.net/users/16750 | Why "Classification" of 4 manifolds is NOT possible? | I'm guessing that you heard this from someone whose reasoning goes "Every finite presentation of a group can be made to give the $\pi\_1$ of a smooth 4-manifold. If we could put any 4-manifold into the Magic List of All, then we could recognize presentations of the trivial group. But no algorithm can do that."
Often ... | 31 | https://mathoverflow.net/users/391 | 73819 | 44,917 |
https://mathoverflow.net/questions/73647 | 8 | Let $\; X\_0,X\_1,X\_2,X\_3,...\;$ be independent and identically distributed (real-valued) random variables.
1.
Suppose $\frac1n \cdot\sum\limits\_{m=0}^n X\_m$ converges in probability. Does it follow that $\operatorname{E}(X\_0)$ exists?
2.
Suppose $\operatorname{E}(X\_0) = 0$ and that $\frac1{\sqrt n} \cdot\sum... | https://mathoverflow.net/users/nan | Do the converses of [weak law of large numbers / central limit theorem] hold? | (As suggested, I promote my comment to an answer, with pgassiat's complement.)
Necessary and sufficient conditions (in terms close to those you want) for the WLLN and the CLT can be found, e.g., in "Foundations of modern probability" by Kallenberg (Theorems 4.16 and 4.17 in the first edition, Theorems 5.16 and 5.17 i... | 8 | https://mathoverflow.net/users/5709 | 73828 | 44,923 |
https://mathoverflow.net/questions/73791 | 3 | Let's work on a Riemannian manifold $M$ of nonpositive sectional curvature.
Fix a unit-speed geodesic $\beta$, and a Jacobi field $\eta$ over it. Assume that $\eta(0)$ is nonzero and orthogonal to $\beta'(0)$, and that $\eta'(0)$ (i.e. $\nabla\_{\beta'} \eta (0)$) equals $0$.
Under these conditions, it's known th... | https://mathoverflow.net/users/1516 | About Jacobi fields on nonpositive curvature | I think that the answer to quetion 2 is yes. The basic property of Jacobi fields defined along a geodesic $\gamma$ (parameterized at speed $1$) is that they satisfy the equation
$$ \eta''(t)=\pm R(\gamma',\eta)\gamma'~, $$
where $R$ is the curvature operator, $\eta''$ is with respect to the covariant derivative along $... | 4 | https://mathoverflow.net/users/9890 | 73834 | 44,927 |
https://mathoverflow.net/questions/73824 | 0 | I am looking for the reference where I can find the proof of the following:
If $A$ is an abelian variety then its tangent bundle is trivial.
| https://mathoverflow.net/users/17420 | Abelian Variety and Tangent Bundle ----Reference Request | See also
<http://www-fourier.ujf-grenoble.fr/~mbrion/notes_bremen.ps>
which proves a converse (corr. 2.3)
| 5 | https://mathoverflow.net/users/11142 | 73846 | 44,935 |
https://mathoverflow.net/questions/73028 | 7 | What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$?
Do we have to calculate $A^{-1}b$, or is this not necessary?
edit: I forgot to mention that A is symmetric and positive definite and sparse (so usually you'd use the conjugate gradient method).
What I have is a convex quadratic $x^TAx... | https://mathoverflow.net/users/6210 | Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently | This is possibly an answer from a practical point of view: If you use the CG method for solving $x=A^{-1}b$ then $b^T A^{-1}b$ can be obtained along the way. However, it has been shown that computing $b^T A^{-1}b$ during the iteration can converge faster than first solving for $x$ and then multiplying $b^T x$. See "Z. ... | 6 | https://mathoverflow.net/users/14039 | 73847 | 44,936 |
https://mathoverflow.net/questions/73729 | 4 | The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B\_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll stick to this case; clearly we could work with other Lie groups as well). The WRT Hilbert space $\mathcal H\_n$ associated... | https://mathoverflow.net/users/35353 | How does one relate the monodromy of the KZ equations with the WRT representation of the braid group? | What I will say is just for quantum $SL(2)$ with $V$ being the 2-dimensional representation (or the analogous object in the representation category). Let us denote by $H$ the quantum group. For $q$ generic you can just use the following: $V^{\otimes n}\cong\bigoplus\_i Hom(V\_i,V^{\otimes n})\otimes V\_i$ as an $H$ mod... | 5 | https://mathoverflow.net/users/6355 | 73848 | 44,937 |
https://mathoverflow.net/questions/73648 | 3 | Probably this is wildly known, but the closest I found was a surface with additional restrictive conditions.
A perfect cuboid is a cuboid having integer side lengths, integer face diagonals and an integer space diagonal leading to positive integer solutions of:
$$a^2+b^2=s\_1^2\qquad \qquad \rm(1)$$
$$a^2+c^2=s\_2^... | https://mathoverflow.net/users/12481 | Does this surface contain all perfect cuboids? | Looks OK to me (the approach anyway - I didn't check the numerator calculation). Ruslan Sharipov also found an explicit equation for the perfect cuboid surface, in a recent ArXiv paper at <http://arxiv.org/abs/1104.1716>. His derivation was much more intricate than yours, but the result looks very similar!
This surfa... | 2 | https://mathoverflow.net/users/10454 | 73865 | 44,947 |
https://mathoverflow.net/questions/60376 | 8 | Even if the answer is no, I am interested in a more specific question.
Let $\Sigma$ be a set of operations of finite arity, $E$ be a set of equations over $\Sigma$ and $\mathcal{A}(\Sigma,E)$ be the respective category of algebras and algebra morphisms. Also denote the free algebra functor by $F: \mathsf{Set} \to \ma... | https://mathoverflow.net/users/5152 | Do coproducts in categories of algebras preserve monos? | Well, I have a counterexample.
Let $\Sigma$ contain two unary operations $a$ and $b$.
Further suppose $E$ contains the equations $a(p) = a(q)$ and $b(p) = b(q)$.
Then $F0 = \emptyset$ because $\Sigma$ contains no nullary operations and we also have the terminal algebra $\mathsf{1}$ with singleton carrier $\{\*\}... | 8 | https://mathoverflow.net/users/5152 | 73870 | 44,950 |
https://mathoverflow.net/questions/73872 | 5 | If $\text{ZF}$ is consistent, then it is not finitely axiomatizable. For if $\Gamma$ is a finite axiomatization, then $\text{ZF}$ proves by reflection that $\Gamma$ has a set model, and hence (since $\Gamma$ axiomatizes $\text{ZF}$) so does $\Gamma$. By the Second Incompleteness Theorem, $\Gamma$ is inconsistent. This ... | https://mathoverflow.net/users/8547 | Does ZF prove that a finite subtheory axiomatizes it over transitive proper class models? | This is a very interesting question, whose answer I find to be quite a subtle but important point. The answer is that it is not possible to formalize the statement in your question in first-order set theory.
To see this, observe first that the theorem you state is actually merely a theorem scheme, asserting of any ax... | 7 | https://mathoverflow.net/users/1946 | 73875 | 44,952 |
https://mathoverflow.net/questions/73889 | 12 | For any space $X$, the first Steenrod square cohomology operation
$$Sq^1\colon H^\ast(X;\mathbb{Z}\_2)\to H^{\ast +1}(X;\mathbb{Z}\_2)$$
is a derivation, meaning that $Sq^1\circ Sq^1 = 0$ and $Sq^1(a\cup b) = Sq^1(a)\cup b + a\cup Sq^1(b)$ (there are no signs since we are working in characteristic two).
Hence we may... | https://mathoverflow.net/users/8103 | $Sq^1$ cohomology of spaces | I think the easiest way to understand the Bockstein spectral sequence is through the exact couple coming from the long exact sequence of cohomology associated to $0\to\mathbb Z\to\mathbb Z\to \mathbb Z/2\to0$. This shows first that indeed the first differential is $Sq^1$ and tells you that the next page is the direct s... | 15 | https://mathoverflow.net/users/4008 | 73891 | 44,960 |
https://mathoverflow.net/questions/73877 | 14 | Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and [this question](https://mathoverflow.net/questions/16342/motivation-for-the-covariant-model-structure-on-sset-s)), on $\mathbf{SSet}/S$ such that:
1. The cofibrations are the monomorphisms.
2. ... | https://mathoverflow.net/users/344 | The weak equivalences in the covariant model structure | Maybe it would be helpful to think about the analogous situation in ordinary category theory. Suppose you are given a category $\mathcal{E}$ and a functor $F$ from
$\mathcal{E}$ to the category of sets. There are several ways to encode this functor:
$(a)$: Via the Grothendieck construction, $F$ determines a category ... | 20 | https://mathoverflow.net/users/7721 | 73893 | 44,961 |
https://mathoverflow.net/questions/73478 | 12 | A variety is $\mathbb{Q}$-factorial if every global Weil divisor is $\mathbb{Q}$-Cartier. How bad singularities are allowed so that the algebraic variety is still $\mathbb{Q}$-factorial? Is a singular curve $\mathbb{Q}$-factorial? For example is a nodal-cuspidal plane curve $\mathbb{Q}$-factorial?
| https://mathoverflow.net/users/2348 | When is an algebraic variety $\mathbb{Q}$-factorial? | I saw this a while ago, and I assumed that someone would give you an answer, but it doesn't
look like anyone will. So I'll make an attempt.
As Ulrich suggests, it is better to stick to
normal varieties, so your nodal/cuspidal curves are immediately eliminated. So we should move to dimension at least two.
A very sim... | 10 | https://mathoverflow.net/users/4144 | 73895 | 44,962 |
https://mathoverflow.net/questions/73699 | 3 | Does someone have examples of extensions of results from convergence theory for riemannian geometry to a lorentzian setting. (I am familiar with the work of M.T.Anderson and co. in CMC gauge, i would like to have other references if possible). Thanks !
| https://mathoverflow.net/users/17388 | convergence theory -> lorentzian geometry | I just noticed this question about Lorentzian convergence for which some results have been obtained, albeit by far not as strong as in the metric case. In case anyone is interested in having a discussion about this, getting some references or some idea as to why it is somewhat more difficult than the standard metric th... | 3 | https://mathoverflow.net/users/17445 | 73901 | 44,965 |
https://mathoverflow.net/questions/73885 | 6 | Let $V$ be a real vector space. A bilinear form $\langle \rangle:V\times V\to {\mathbb{R}}$ induces a linear functional $\theta$ on the tensor product $V\otimes V$ given by sending the finite sum $\sum\_i v\_i\otimes w\_i $ to $\sum\_i \langle v\_i,w\_i\rangle$.
Is there a name for this induced linear functional?
I... | https://mathoverflow.net/users/nan | Is there a name for this map induced by bilinear forms? | The correspondence you describe is part of the definition of the tensor product: $V \otimes W$ is defined to have the universal property that for any $U$, we have $$\operatorname{Hom}(V \otimes W, U) = \operatorname{Bil}(V \times W, U).$$ I wouldn't even give it a different name: the bilinear form is the same as the ma... | 4 | https://mathoverflow.net/users/6545 | 73908 | 44,969 |
https://mathoverflow.net/questions/73923 | 9 | Let $V$ be a real vector space, and let $(\cdot,\cdot;\cdot,\cdot) : V^4 \to \mathbb{R}$ be a multilinear form with the following properties:
1. $(x,y;z,w) = (y,x;z,w) = (x,y;w,z)$ (symmetry in the first and second pairs)
2. $(x,x;z,z) \ge 0$ (positive semidefiniteness in the first and second pairs).
>
> Must suc... | https://mathoverflow.net/users/4832 | Multilinear generalization of Cauchy-Schwarz inequality | Even the inequality $(x,z;x,z)^2 \le (x,x;z,z)(z,z;x,x)$ is false:
Let $V = \mathbb{R}^2$, with basis $x,z$. Take $(x,x;x,x) = 100$, $(x,z;x,x)=0$, $(z,z;x,x)=1$, $(x,x;x,z)=0$, $(x,z;x,z)=50$, $(z,z;x,z)=0$, $(x,x;z,z) = 1$, $(x,z;z,z)=0$, $(z,z;z,z)=100$, and extend to all of $V^4$ by symmetry and multilinearity.
... | 9 | https://mathoverflow.net/users/2363 | 73934 | 44,981 |
https://mathoverflow.net/questions/73936 | 2 | Let $X$ be a scheme. $U$ is an open subscheme of $X$. Assume $f$ is a global section on $X$ which is not a zero divisor, then the restriction of $f$ to $U$ is still a non-zero divisor?
If $X$ is affine, the answer is obvious true. I don't know the answer for a general scheme.
This is a question raised in the defini... | https://mathoverflow.net/users/3525 | The restriction of a global section which is not a zero divisor is still a non-zero divisor? | Here's a counterexample.
Let $P=\mathbb{P}^1$, $X=\mathbb{A}^1$, and attach $X$ to $P$ along a single point $\{x\}$. Then there is a global section $f$ which is nonzero on $X$ except at $x$, and is identically zero on $P$. Moreover, $f$ restricts to a zero divisor on the open subvariety $X\cup P/\lbrace y\rbrace$, wh... | 6 | https://mathoverflow.net/users/5513 | 73946 | 44,987 |
https://mathoverflow.net/questions/73940 | 15 | In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, [here](https://mathoverflow.net/questions/10481/when-is-a-classification-problem-wild/10484#10484)). In logic, there is a notion of an [undecidable problem](http:/... | https://mathoverflow.net/users/9672 | Are wild problems related to undecidable ones? | Yes, there is a connection, but I think it is conjectural in its full generality. The mosst general reference could be, where it is proven, that for a subclass of wild algebras, the representation theory is undecidable:
Mike Prest: Wild representation type and undecidability, Comm. Alg. 19 (3), 1991.
It is also wel... | 18 | https://mathoverflow.net/users/15887 | 73947 | 44,988 |
https://mathoverflow.net/questions/73942 | 4 | Let $S$ be an infinite graph, $G$ is a group acting (effectively) on $S$ with finite quotient graph $S/G$. Make $S/G$ into graph of groups in obvious way by assigning stabilizers at vertices and edges.
Let $\tilde{S}$ be universal cover of $S$ and $H$ be a group acting (effectively) on $\tilde{S}$ with same quotient ... | https://mathoverflow.net/users/17456 | Groups acting on graph | If I understand your question correctly then the answer is 'no'. Indeed, whenever countable $\Gamma=A\*\_C B$ with $|A:C|=|B:C|=\infty$ then the Bass--Serre tree $S$ is a countably branching regular tree and the quotient graph $S/\Gamma$ is just a single edge. So, taking $S=\widetilde{S}$ and $K=G$, if the answer to yo... | 5 | https://mathoverflow.net/users/1463 | 73948 | 44,989 |
https://mathoverflow.net/questions/73951 | 1 | Where can I find references that discuss important classes of Infinite Hopf Algebras. By important classes, I mean heavily used in research and of relevance to Hopf Algebraist(s),Physicists, Analysts(Real/Complex),..etc.
| https://mathoverflow.net/users/17265 | References For Important Hopf Algebras | [Shahn Majid](http://www.maths.qmw.ac.uk/~majid/Welcome.html)'s book *Foundations of quantum group theory* (Cambridge Univ. Press 1995, 2000) has lots of examples and of classes of examples. These are not only examples of quantum groups in the narrow sense (cf. the $n$Lab [page](http://www.ncatlab.org/nlab/show/quantum... | 4 | https://mathoverflow.net/users/35833 | 73953 | 44,992 |
https://mathoverflow.net/questions/73914 | 5 | I don't know if this question is appropriate to this site, but I posted [here](https://math.stackexchange.com/questions/58568/on-a-remark-in-foundations-of-mechanics-2nd-edition-by-abraham-and-marsden) without an answer, so I tried this alternative.
Given a $2$-form $\omega$ on a manifold $M$, let us denote by $N$ th... | https://mathoverflow.net/users/12617 | On a remark in Foundations of mechanics, 2nd Edition, by Abraham and Marsden | The answer is 'no'. To see why, just take any nondegenerate $2$-form $\omega$ on, say, $\mathbb{R}^6$, that has the property that $d\omega$ is not a multiple of $\omega$. (This will be true for a generic such $2$-form.) The kernel of this $\omega$ is trivial, but now, you can just regard it as being defined on $\mathbb... | 5 | https://mathoverflow.net/users/13972 | 73961 | 44,996 |
https://mathoverflow.net/questions/73977 | 2 | Let $A \subset \mathbb N$ be a antichain with respect to divisibility. Does this imply that the density of $A$ is $0$?
| https://mathoverflow.net/users/14233 | Has every divisibility-antichain density zero? | Such a subset is called "primitive"; such a set must have lower density zero but (as shown by Besicovitch) does not need to have asymptotic density zero. Check out the references and results of this recent paper of Greg Martin and Carl Pomerance:
<http://arxiv.org/pdf/1009.1014v2>
| 8 | https://mathoverflow.net/users/17465 | 73981 | 45,004 |
https://mathoverflow.net/questions/73979 | 0 | Let $H$ be a Hilbert space and $H'\le H$ a subspace as Hilbert spaces (I mean, the inner product in $H'$ is the same inner product of $H$ restricted to $H'$).
If we take $f:H\to H$ an automorphism of Hilbert spaces which fixes $H'$ pointwise, notice that if $f(a)=b$ then $\|a\|=\|b\|$ and $dist(a,H')=dist(b,H')$. Is ... | https://mathoverflow.net/users/17464 | Automorphisms in Hilbert spaces | No, because the projections of $a$ and $b$ to $H'$ might be different.
| 3 | https://mathoverflow.net/users/6794 | 73985 | 45,005 |
https://mathoverflow.net/questions/73922 | 14 | Note: by fixed points, I always mean homotopy fixed points.
As explained in Jacob Lurie's [paper](http://www.math.harvard.edu/~lurie/papers/cobordism.pdf) on the cobordism hypothesis, we have an action of O(2) on the $\infty $-groupoid $X$ given by considering fully dualizable objects and invertable morphisms in some... | https://mathoverflow.net/users/7762 | Homotopy Fixed Points of SO(2) on Fully Dualizable Algebras | I might be confused about your question. Are you asking...
1. How is trivializing the $O(n)$-action the same as giving an $O(n)$-equivariant non-degenerate trace? (as per Lurie's theorem 3.1.8).
2. How can we identify the $SO(2)$-action with the usual $SO(2)$-action on Hochschild homology?
For the first one, I thin... | 10 | https://mathoverflow.net/users/184 | 73986 | 45,006 |
https://mathoverflow.net/questions/73931 | 7 | I have finite set of geolocation point data, and I'd like to estimate the fractal dimension. I know there are several ways to do this, and some of them give different numbers. What is the most appropriate fractal dimension to look at and what method do you recommend I use to estimate it numerically?
Thanks
| https://mathoverflow.net/users/942 | Estimating the fractal dimension of a point cloud | It depends what you want to measure. For real-life data box-counting dimension based on [Renyi entropy](http://en.wikipedia.org/wiki/R%C3%A9nyi_entropy) (of order $q$) is a common choice. For some problems $q=1$ (Shannon entropy) or $q=2$ (collision entropy) may be privileged. You can plot fractal dimension for any $q$... | 4 | https://mathoverflow.net/users/9093 | 73988 | 45,008 |
https://mathoverflow.net/questions/73960 | 3 | Let $R$ be a discrete valuation ring qith quotient field $Q$ and let $t\in R$ be a generator of the unique maximal ideal in $R$. Let $V$ be a finite-dimensional $Q$-vector space. Then one can consider the set of all homothety classes of $R$-lattices in $Q$ (i.e. finitely generated $R$-submodules of the same rank).
T... | https://mathoverflow.net/users/3969 | Is this the CAT(0) metric on an affine building? | The important thing seems to be that one needs to understand the connection between the CAT(0)-realization of the Coxeter complex that corresponds to the Weyl group and the description of an apartment given in terms of lattices. Since a geometric realization of this Coxeter complex will basically describe a geometric r... | 7 | https://mathoverflow.net/users/12824 | 73990 | 45,009 |
https://mathoverflow.net/questions/73972 | 1 | What is known about local structure of actions of semi-simple groups? More precisely, suppose I have a semi-simple group $G$ acting on a variety $V$, and $x\in V$. Assume that the stabilizer of $x$ is a parabolic subgroup $P\subset G$. Can I always find a slice $x\in V'\subset V$ such that the natural map $G\times\_P V... | https://mathoverflow.net/users/6772 | Linearization of actions of semi-simple groups | [Edit]: my previous counterexample was irredeemably wrong; hopefully this one works.
Suppose that there exists an étale $G$-equivariant map $V' \times^{P}G \to V$; then there exists an invariant neighborhood $U$ of $x$ in $V$ (the image of this map) such that the connected component of the identity in the stabilizer ... | 2 | https://mathoverflow.net/users/4790 | 73994 | 45,012 |
https://mathoverflow.net/questions/73959 | 4 | Let $X,Y,Z$ be reduced algebraic varieties, and let $Y$ and $Z$ be normal. Let $f:X \to Y$ and $g:X \to Z$ two surjective projective morphisms of algebraic varieties such that the geometric fibers of $f$ and $g$ coincide. Is there an isomorphism $h:Y\to Z$ such that $g=h \circ f$?
| https://mathoverflow.net/users/4096 | do geometric fibers determine scheme-theoretic image? | In positive characteristic, you get a counterexample by taking $X=Y=Z=$ the affine line (say), $f$ the identity and $g$ the Frobenius map.
Assume now that the ground field is algebraically closed of characteristic zero. Consider the map $(f,g):X\to Y\times Z$. Its image $\Gamma$ is a closed subvariety of $Y\times Z$.... | 11 | https://mathoverflow.net/users/7666 | 73995 | 45,013 |
https://mathoverflow.net/questions/73996 | 4 | Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
| https://mathoverflow.net/users/17164 | Hausdorff dimension of graphs . | The graph of any Lipschitz function $f\colon [a,b]\to\mathbb{R}$ has Hausdorff dimension $1$ (this follows since Hausdorff dimension is invariant under bi-Lipschitz mappings). Your example of $f(x) = \sin(1/x)$ also has a graph with Hausdorff dimension $1$, since the graph can be decomposed into a countable union of cu... | 11 | https://mathoverflow.net/users/5701 | 73997 | 45,014 |
https://mathoverflow.net/questions/73952 | 4 | (This is based on [my earlier question](https://mathoverflow.net/questions/73815/does-zf-prove-that-topological-groups-are-completely-regular), but I think this one would be easier to answer.)
Let $\langle X,\mathbf{\delta} \hspace{.01 in} \rangle$ be a [separated proximity space](http://en.wikipedia.org/wiki... | https://mathoverflow.net/users/nan | Does ZF prove that proximity spaces are completely regular? | The answer is no. In fact, it is consistent with ZF that $(\*)$ there exists an infinite compact Hausdorff space $X$ such that every continuous function $f\colon X\to\mathbb R$ is constant, so that $X$ is not even completely Hausdorff.
A simple example can be given using a Fraenkel–Mostowski permutation model of ZFA ... | 6 | https://mathoverflow.net/users/12705 | 73999 | 45,015 |
https://mathoverflow.net/questions/74004 | 23 | I was tutoring someone in analysis and realized I have no idea where this notation comes from (or analogous terms: σ-additive, σ-ring, etc). I would like to know why the letter σ was chosen. I can't think of anything relevant that starts with "S" in either English or French. My German is nearly nonexistent, but I didn'... | https://mathoverflow.net/users/4087 | What does the σ in σ-algebra stand for? | From [Elstrodt's book](http://books.google.com/books?id=WECVEDeljqgC&printsec=frontcover&dq=elstrodt+integrationstheorie&hl=de#v=onepage&q&f=false) *Maß- und Integrationstheorie*, pages 13-14:
>
> Bei den Wörtern „$\sigma$-Ring", „$\sigma$-Algebra" weist der Vorsatz „$\sigma$-..." darauf hin, daß das betr.
> Meng... | 44 | https://mathoverflow.net/users/450 | 74013 | 45,021 |
https://mathoverflow.net/questions/73984 | 8 | While investigating certain conformal blocks line bundles on $\overline{M}\_{0,n}$, I was led to what seems to be an identification between two spaces of invariants, and I am curious if there is a direct way to see this identification.
**Statement**: for any integers $n\ge 4$ and $r\ge 2$, and any integers $i\_1,\ldo... | https://mathoverflow.net/users/10930 | Classical invariants involving exterior powers of standard representation | To follow up on Sasha's answer, yes there is a natural isomorphism of vector spaces which lifts the combinatorial equality. All isomorphisms in this answer will be natural.
**Schur-Weyl duality:**
Let $\lambda$ be a partion; set $d = |\lambda|$ and let $m$ be greater than or equal to the number of parts of $\lambd... | 12 | https://mathoverflow.net/users/297 | 74024 | 45,027 |
https://mathoverflow.net/questions/74035 | 2 | I am trying to find some work done on the following:
$$\sum\_{d \vert n}\frac{2^{\omega(d)}}{d}\mu(d)$$
where $\omega(d)$ is the number of distinct prime factors of $d$ and $\mu$ is the mobius function. I saw something about
$$\sum\_{d \vert n}\frac{\mu(d)}{d}=\phi(n)/n$$
(where $\phi$ is the Euler phi function) on pl... | https://mathoverflow.net/users/10920 | Sum of Mobius function and omega function | Whenever $f(n)$ is a multiplicative function, so is $g(n) = \sum\_{d\mid n} f(d)$. Therefore to evaluate your function, you only need to know its values on prime powers. Since
$$
\sum\_{d\mid p^k} \frac{2^{\omega(d)}}d \mu(d) = \sum\_{j=0}^k \frac{2^{\omega(p^j)}}{p^j} \mu(p^j) = 1 - \frac2p,
$$
it follows that
$$
\sum... | 5 | https://mathoverflow.net/users/5091 | 74039 | 45,036 |
https://mathoverflow.net/questions/74036 | 3 | If $G$ is a group and $H$ is a subgroup of $S\_n$ we can form their wreath product $G \wr H = \{(g\_1, ..., g\_n; \pi): g\_i \in G$ and $\pi \in H\}$. I'm wondering whether the following is correct:
1. $<(e, ..., g\_i, ..., e; e)> = \{(g\_1, ..., g\_n; e)\}$
2. $\{(g\_1, ..., g\_n; e)\} \circ \{(e, ..., e; \pi)\} = G... | https://mathoverflow.net/users/nan | How to efficiently generate a wreath product? | Note that the wreath product is a semidirect product with normal subgroup $G^n$ and complement $H$. A group theory book that defines wreath product will cover this fact.
Part 1. seems to ask whether the normal $G^n$ is generated by elements of the form $(e, \ldots, g\_i, \ldots,e)$, i.e. those which are the identity ... | 3 | https://mathoverflow.net/users/16886 | 74040 | 45,037 |
https://mathoverflow.net/questions/74009 | 1 | I have a very simple question, because I basically just need to know if a certain train of thought I've had is correct. My reference is Liu's book "Algebraic Geometry and Arithmetic Curves", in particular Proposition 8.1.15, and of course Hartshorne. Consider the following situation:
Let $f:W\to X$ be a morphism of l... | https://mathoverflow.net/users/9947 | Functoriality of the Blow-Up | Suppose $f={\rm Id}\_X$, $X={\bf A}^3\_{\bf C}$ (affine space of dimension $3$ over the complex numbers). Suppose that $\cal I$ is the sheaf of ideals of a smooth curve going through $0$ and that $\cal K$ is the sheaf of ideals of the point $0$ in ${\bf A}^3\_{\bf C}$. Then the
pull-back of ${\cal J}={\cal I}$ to $\wi... | 5 | https://mathoverflow.net/users/17308 | 74042 | 45,038 |
https://mathoverflow.net/questions/74045 | 9 | Is there a function $p:\mathbb N\to \{ 1,-1 \} $ and a fixed $N\in \mathbb N$ such that for every $n \geq N$ we get:
$\sum \_{i=0} ^{n} p(i)\binom {n}{i}=0$
?
Obviously $p(i)=(-1)^i$ works for $N=1$, and so does $p(i)=(-1)^{i+1}$, but are there any others?
(my personal guess is that there aren't)
| https://mathoverflow.net/users/17476 | Zero sum of binomial coefficients | No, there are no others.
In fact, define a function $q : \mathbb N\to\left\lbrace 1,-1\right\rbrace$ by $q\left(i\right) = \left(-1\right)^i p\left(i\right)$ for every $i\in\mathbb N$. Then, $\sum\limits\_{i=0}^n p\left(i\right) \binom{n}{i} = 0$ becomes $\sum\limits\_{i=0}^n \left(-1\right)^i q\left(i\right) \binom{... | 16 | https://mathoverflow.net/users/2530 | 74048 | 45,042 |
https://mathoverflow.net/questions/73556 | 1 | I've derived equations for 2d polygon's moment of inertia using Green's Theorem (constant density \rho)
$$I\_y = \frac{\rho}{12}\sum\_{i=0}^{i=N-1} ( x\_i^2 + x\_i x\_{i+1} + x\_{i+1}^2 ) ( x\_i y\_{i+1} - x\_{i+1} y\_i )$$
$$I\_x = \frac{\rho}{12}\sum\_{i=0}^{i=N-1} ( y\_i^2 + y\_i y\_{i+1} + y\_{i+1}^2 ) ( x\_{i+... | https://mathoverflow.net/users/17362 | Calculating moment of inertia in 2d planar polygon | Sorry for my mistake. both equations were slightly incorrect.
Let me write the correct equations
$$I\_y = \frac{\rho}{12}\sum\_{i=0}^{i=N-1} ( x\_i^2 + x\_i x\_{i+1} + x\_{i+1}^2 ) ( x\_i y\_{i+1} - x\_{i+1} y\_i )$$
$$I\_x = \frac{\rho}{12}\sum\_{i=0}^{i=N-1} ( y\_i^2 + y\_i y\_{i+1} + y\_{i+1}^2 ) ( x\_i y\_{i+1}... | 1 | https://mathoverflow.net/users/17362 | 74055 | 45,046 |
https://mathoverflow.net/questions/71955 | 6 | The fundamental braid $\Delta\_n \in B\_n$ is simply a twist by $\pi$ applied to the entire row of $n$ strands. In terms of Artin generators, it is given by
$$
\Delta\_n = (\sigma\_1 \sigma\_2 \cdots \sigma\_{n-1})(\sigma\_1 \sigma\_2 \cdots \sigma\_{n-2})\cdots (\sigma\_1 \sigma\_2) \sigma\_1~.
$$
The square of $\Delt... | https://mathoverflow.net/users/12695 | Jones Polynomial of the trace closure of the fundamental braid | Calculation of the Jones polynomial of this link is a (good) exercise in representation
theory. As you have observed by Schur's lemma, in any irreducible representation it is, up to a scalar,
a square root of the identity. This scalar can be obtained by a determinant argument.
So we reduce to the situation where the ei... | 18 | https://mathoverflow.net/users/5973 | 74062 | 45,049 |
https://mathoverflow.net/questions/73863 | 13 | Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all 1's. This problem looks vaguely like a semidefinite programming problem, except that both the matrix $(M+D)$ and it's in... | https://mathoverflow.net/users/14424 | Seeking proof for linear algebra constraint problem. | (Edit: my original answer was perhaps not clear enough, let me try to improve it).
First some notation: for a matrix $x$, let me denote by $E(x)$ the diagonal matrix with the same diagonal as $x$: if $x=(x\_{i,j})\_{i,j\leq n}$, $E(x) = (x\_{i,j}\delta\_{i,j})\_{i,j \leq n}$. Equivalently, $E$ is the orthogonal proje... | 5 | https://mathoverflow.net/users/10265 | 74067 | 45,051 |
https://mathoverflow.net/questions/74061 | 1 | Does there exist a conformal smooth extension of a smooth function? Smooth extension is guaranteed by Whitney extension theorem. does that theorem also says for conformality.
Precisely the question is the following:
Let $ D= \{z: |z|\leq 1,I[z]\geq 0 \}$. Define $f\colon D \to\mathbb R^2$ such that all the derivative... | https://mathoverflow.net/users/16031 | Conformal extension | Firstly, conformal is the same as complex-analytic, and it is much more convenient to think in those terms.
Secondly, there are well-known examples of functions such as $f(z)=\sum\_{n=0}^\infty z^{n!}$ that are analytic on the open unit disc but diverge at a dense set of points on the unit circle so cannot be continu... | 3 | https://mathoverflow.net/users/10366 | 74075 | 45,054 |
https://mathoverflow.net/questions/74032 | 4 | If $X$ is an infinite graph, $G$ is a group acting on $X$ with finite quotient; make $Y=X/G$ into graph of groups by attaching stabilizers at vertices and edges. Let $Z$ be a graph of groups, with graph equal to graph of $X/G$, all groups at vertices and edged being finite(but not all trivial), and suppose there is mor... | https://mathoverflow.net/users/17456 | Finite quotients of graphs | No. For instance, let $G=\mathbb{Z}/4\*\mathbb{Z}/4$ and let $X$ be the Bass--Serre tree, the infinite 4-regular tree. Now let $Z$ be the graph of groups corresponding to $\mathbb{Z}/2\*\mathbb{Z}/2$. Then the Bass--Serre tree for $Z$ is just a line. But if $Z$ were a quotient of $X$ then it would follow that the line ... | 3 | https://mathoverflow.net/users/1463 | 74077 | 45,056 |
https://mathoverflow.net/questions/74010 | 2 | Edit: the original question was imprecise, I'm sorry about that. I hope this is better.
Let $k$ be a field, $S/k$ any scheme, $G/k$ be an algebraic group and $X$ an $S$-torsor under $G$ (so $G$ acts simply transitively on the fibers of $G \to S$). Under what (sufficient/necessary/whatever) conditions on the schemes $... | https://mathoverflow.net/users/1107 | (corrected) When does a torsor trivialize over the algebraic closure? | -The following is a general condition for trivialization:
Let $\pi:X\to S$ be the structure map and let $T\_G$ be the sheaf of trivialization maps where for $U\subset X$ open we have $T\_G(U) := \lbrace \psi: \pi^{-1}(U)\to U\times G, \mbox{an isomorphism such that } \pi = p\_1\circ\psi\rbrace$ (you can replace $G$ w... | 1 | https://mathoverflow.net/users/1467 | 74079 | 45,058 |
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