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https://mathoverflow.net/questions/69810 | 8 | Is there a sequence of connected finite-dimensional subgroups Gi of the circle diffeomorphism group G with the following properities:
(a) Gi is contained in Gj for i < j
(b) The union of Gi is dense in G
More rigorously "finite dimensional subgroup of circle diffeomorphism group" means a Lie group H with smooth f... | https://mathoverflow.net/users/11146 | Finite-dimensional subgroups of circle diffeomorphism group | If $G$ is a connected Lie group acting transitively and faithfully on a connected smooth $1$-manifold, then $G$ is at most $3$-dimensional; in fact its Lie algebra embeds in that of $SL\_2(\mathbb R)$. (Edit: Alex Eskin's answer says this in detail, with a reference.)
Each orbit of an action of a connected topologica... | 8 | https://mathoverflow.net/users/6666 | 69841 | 42,835 |
https://mathoverflow.net/questions/69843 | 0 | Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v\_{n}: n\ge 1$} is an infinite linearly independent subset of $M$ that is bounded and bounded away from zero (in the case I'm considering, the collection {$v\_... | https://mathoverflow.net/users/16008 | When are operators extended by linearity bounded? | The hypotheses you gave are not sufficient to make your proposed operator $S$ bounded. You could have $v\_n$ very close to $v\_m$ while $v\_{n+1}$ is far from $v\_{m+1}$, and this could happen repeatedly with greater and greater discrepancies between the two distances.
| 0 | https://mathoverflow.net/users/6794 | 69845 | 42,837 |
https://mathoverflow.net/questions/69773 | 9 | In the book "[The Mathematical Experience](http://books.google.com/books?id=lMdz84dWNnAC&q=Doob#v=snippet&q=Doob&f=falseBlockquoteblah)" it says:
>
> "An infinite [binary] sequence $x\_1, x\_2, \ldots$ is called random in the sense of von Mises if every infinite sequence $x\_{n\_1}, x\_{n\_2}, \ldots$ extracted fr... | https://mathoverflow.net/users/4692 | Did Joseph Doob prove that random sequences don't exist? | At Gerald Edgar's suggestion, I promote my comment to an answer.
There is a good discussion of the questions raised here in the chapter on randomness in Seminumerical Algorithms, Volume 2 of Knuth's The Art Of Computer Programming.
| 10 | https://mathoverflow.net/users/3684 | 69851 | 42,842 |
https://mathoverflow.net/questions/69807 | 4 | I came across a Chinese reference in the paper "On the spectral radius of trees with fixed diameter" by Guo and Shao. The attribute the following to Q. Li, K.Q. Feng in: "On the largest eigenvalue of graphs", Acta Math. Appl. Sinica 2 (1979) 167–175.
Let $\lambda\_1(H)$ be the spectral radius of the adjacency matrix ... | https://mathoverflow.net/users/6209 | Spectral radius of a proper subgraph | Here is a simple proof. Without loss of generality, $G'$ is obtained from $G$
by deleting some edges (and keeping all vertices). Let $A$ and $A'$ denote
the adjacency matrices of $G$ and $G'$, respectively, and let $x'$ be an
eigenvector of $A'$, belonging to the eigenvalue $\lambda\_1(G')$, such that
all coordinates o... | 8 | https://mathoverflow.net/users/9924 | 69855 | 42,844 |
https://mathoverflow.net/questions/69242 | 3 | For positive real $x\_1$ , $x\_2$ ,..., define their $k$th partial harmonic mean as $h\_k = k/(1/x\_1 +\cdots+1/x\_k)$ for $k = 1, 2, ...,$ and let
$\alpha=\sup\_{x\_1,x\_2,... \geqslant0}\: \lim\_{n\rightarrow\infty}\dfrac{h\_1+\cdots+h\_n}{x\_1+\cdots+x\_n}.$
What is this bound, and for which $x\_1$ , $x\_2$ ,...... | https://mathoverflow.net/users/7458 | What bounds the ratio of summed partial harmonic means to a sum? | This one is an old classic. I think it is due to Hardy, but you should check Hardy-Littlewood-Polya's "Inequalities". This is equivalent to [Hardy's inequality](http://en.wikipedia.org/wiki/Hardy%27s_inequality) with $p=-1$. I think Hardy claimed the result for all $p>1$, and the observation that it holds for all $p\le... | 5 | https://mathoverflow.net/users/2384 | 69859 | 42,845 |
https://mathoverflow.net/questions/69858 | 4 | Suppose I have a homogeneous cubic polynomial $f(w,x,y,z)$ and I let $X$ be the set of points in $\mathbb{R}^4$ where $f(w,x,y,z)=0$ and $w^2+x^2+y^2+z^2=1$. Suppose that this is a smooth surface, and I give it the metric inherited from $\mathbb{R}^4$. Does anyone know a nice formula for the curvature of $X$ in terms o... | https://mathoverflow.net/users/10366 | Curvature formula | There is basic formula in Riemannian geometry that, used twice, gives fairly direct answer. Let $M$ be a Riemannian manifold, let $N$ be a codimension $1$ submanifold of $M$, and let $f$ be a function defined on $M$. Call $f$ also the restriction of $f$ to $N$. Then the Hessian of $f$ on $N$ and the restriction to $TN$... | 10 | https://mathoverflow.net/users/9890 | 69867 | 42,851 |
https://mathoverflow.net/questions/69863 | 2 | Let $C$ be a symmetric monoidal category. Fix an object $X$, let $S$ denote the symmetry $X \otimes X \to X \otimes X$. Also define $X^{\otimes n}$ by induction on $n$: $X^{\otimes 0} = 1$, $X^{\otimes (n+1)} = X^{\otimes n} \otimes X$. Now it is "absolutely clear" that we have a canonical action of the symmetric group... | https://mathoverflow.net/users/2841 | Action on tensor power and "element notation" in monoidal categories | There's a point of view on symmetric monoidal categories (dating back to Graeme Segal's "Categories and cohomology theories" and then taken up by Bertrand To\"en in "Dualit\'e de Tannaka superieure, I: Structures monoidales" to define symmetric monoidal $n$-categories) which stresses the symmetric group action from the... | 3 | https://mathoverflow.net/users/8320 | 69868 | 42,852 |
https://mathoverflow.net/questions/69846 | 11 | I've been told that when applying for a teaching position, your reference letters can be written by anyone who is familiar with your teaching capabilities in detail. I feel that this primarily just means students, but I wonder if there's some unspoken rule that reference letters should come from people in positions of ... | https://mathoverflow.net/users/7521 | Reference letters for teaching positions | IMO, The best teaching letters come from senior faculty members who have supervised you as a teacher and have sat in on several of your classes to observe your teaching. At many schools, it is standard procedure for this to happen. The letter should also discuss student evaluations, including numerical scores and some ... | 21 | https://mathoverflow.net/users/11926 | 69869 | 42,853 |
https://mathoverflow.net/questions/69857 | 4 | I know this title makes what I am about to ask sound like an off topic CS theory question but please bear with me because I assure you that it is not! (Well mostly, actually I am about ~90% certain that this is a perhaps routine application of representation theory...) If you just want the problem without the backstory... | https://mathoverflow.net/users/4642 | Bits and orbits | I suspect the problem is intended to include some requirement that the coding scheme be efficiently computable, because if one doesn't care at all about efficiency then the following (rather silly) scheme would provide codes of the shortest possible length. By the "first member" of an orbit, I mean the lexicographicall... | 3 | https://mathoverflow.net/users/6794 | 69880 | 42,858 |
https://mathoverflow.net/questions/69802 | 2 | 1. Consider a smooth projective surface $S\subset\Bbb P^N\_{\Bbb C}$ which is a complete intersection of hypersurfaces of degrees $(d\_1,..,d\_{k\ge2})$ with $d\_i\ge2$ for all i. Is it true that for such surfaces $c^2\_1\le 2c\_2$? (i.e. much better than BMY)
At least asymptotically (i.e. for high enough $d\_i$'s)?
... | https://mathoverflow.net/users/2900 | Looking for an inequality between Chern and Todd classes (something in style of Bogomolov-Miyaoka-Yau) | By the formulae in [Barth-Peters-Van de Ven, Chapter V] one has, for a surface which is complete intersection of type $(d\_1, \ldots, d\_{n-2})$ in $\mathbb{P}^n$:
$$c\_1^2(X)= \big(\sum d\_i-(n+1)\big)^2 \prod d\_i,$$
$$c\_2(X)=\bigg[\binom{n+1}{2}-(n+1)\sum d\_i+\sum d\_i^2 +\sum\_{i < j} d\_id\_j \bigg]\prod d\_i.$$... | 5 | https://mathoverflow.net/users/7460 | 69881 | 42,859 |
https://mathoverflow.net/questions/69718 | 2 | Riemann-Siegel's approximate functional equation
$\zeta(s) = \sum\_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum\_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}) $
is the starting point for accurate numerical estimates of $\zeta(s)$ as a function of the partial sums in s an... | https://mathoverflow.net/users/15020 | Riemann-Siegel's approximate functional equation for fixed t and Re(s)≠1/2 | The assumption $2\pi xy=t$ (or something similar) is certainly necessary to guarantee that you get an approximation to the zeta-function and not some other function. For instance, if we choose $x=y=t/2\pi$ then the approximate functional equation "approximates" twice the zeta-function. One can see this as follows:
In... | 6 | https://mathoverflow.net/users/3659 | 69891 | 42,864 |
https://mathoverflow.net/questions/69885 | 6 | Let $C\subset\mathbb{P}^r$ be a smooth nondegenerate curve (not contained in any hyperplane) of degree $d$ genus $g>0$. Consider the tangential variety $X$ of $C$: $X=\cup\_{p\in C}T\_pC\subset \mathbb{P^r}$. This is a surface in $\mathbb{P}^r$ which is singular along $C$. My feeling is that $X$ can not be contained in... | https://mathoverflow.net/users/10646 | quadrics containing the tangential variety of a curve | **Edited.**
Here is a construction of curves $C$ on a four-dimensional quadric $Q^4$ such that $TC\subset Q^4$. I am sure that this is a classical construction, (it might be I saw it previously and forgot).
**Construction.** Recall that $Q^4$ is isomorphic to $G(2,4)$ -- the Grassmanian of $2$-planes in a four-dim... | 5 | https://mathoverflow.net/users/943 | 69894 | 42,866 |
https://mathoverflow.net/questions/69892 | 1 | what is the three parameter family of plane projective transformations which fix a unit circle at the origin(that is map the unit circle to itself)?
I understand that one such transformation is a rotation but that accounts for just one parameter, what are the other two?
| https://mathoverflow.net/users/2705 | projective geometry question | Plane projective transformations can be viewed as linear operators in a 3D vector space V. A circle (more generally an ellipse) in the projective plane P(V) corresponds to a cone C in V. Such a cone is the vanishing locus of an indefinite quadratic form Q on V. Thus the operators we look for are operators preserving Q.... | 4 | https://mathoverflow.net/users/11146 | 69899 | 42,868 |
https://mathoverflow.net/questions/69884 | 3 | Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int\_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
You can assume that $\mu$ is non-negative, i.e., a measure on $R$.
The problem is how $u(t,x)$ behaves for $x$ fixe... | https://mathoverflow.net/users/36814 | Long time behavior of the heat equation on R | Denote $\Gamma(x,t)$ the fundamental solution of the heat equation form the integral. By the theorem of L. Schwarz for any $\mu\in S'(S)$ there is a number $m\in \mathbb N$ and $C>0$ such that $$|u(x,t)|=|(\mu,\Gamma(t,\cdot))|\le C\|\Gamma(t,\cdot)\|\_m,$$ where
$
\|\varphi\|\_m=$
$\sup\_{\alpha \le m,\ x \in \mathb... | 4 | https://mathoverflow.net/users/14551 | 69902 | 42,870 |
https://mathoverflow.net/questions/69853 | 0 | A B are totally-ordered sets and there exist two maps f and g such that f is a order-preserving injection from A to B and g is a order-preserving injection from B to A.
Q: Are A and B necessarily similar?
| https://mathoverflow.net/users/16323 | order-preserving question | In spite of not knowing whether similar means isomorphic or not, the answer seems to be no.
After an idea from Joel Hamkins, I transcribe some of the comments above.
I start the parade with: "No. Although all countable dense total orders without endpoints are isomorphic, there are countable total orders which are n... | 3 | https://mathoverflow.net/users/3402 | 69915 | 42,877 |
https://mathoverflow.net/questions/69879 | 2 | I hope there is a straighforward literature-pointer here.
If I were interested in $\sum\_{t=1}^{n} f(t) X\_{t}$, where $X\_{t}$ consists of independent normal random variables, I could approximate the sum as an Ito integral, and then (if $f(t)$ is reasonably nice) get a good answer for the resulting approximation. Al... | https://mathoverflow.net/users/146 | Stochastic Integrals and Cauchy Variables | Let $\{Y(t):t\ge 0\}$ be a symmetric 1-stable Lévy process. Then $Y$ is a càdlàg process with $E[e^{iuY(t)}]=e^{-t|u|}$. A Levy process is a semimartingale, so we may define the usual stochastic integral with respect to $Y$. Consequently, if $f$ is continuous, then the stochastic integral is the limit in probability of... | 5 | https://mathoverflow.net/users/15575 | 69921 | 42,880 |
https://mathoverflow.net/questions/69904 | 4 | Let $S$ be a compact R.S of genus $\geq 2$. In the paper "Stable and unitary vector bundles on compact Riemann surfaces" (by Narasimhan and Seshadri), they claim that there is a branched covering map from the upper half plane to $S$ which is ramified at exactly one point (with index $N$) (i.e. $S$ is the quotient of $\... | https://mathoverflow.net/users/3709 | Branched covers of compact Riemann surfaces | From your group-theoretic description, it seems to me that you are asking for a covering $S' \to S$ which is only ramified over one point of $S$, and the ramification index of each point of the inverse image divides $N$. So, Felipe's construction should work.
Here is a more direct construction. Suppose that $S$ is co... | 10 | https://mathoverflow.net/users/4790 | 69929 | 42,884 |
https://mathoverflow.net/questions/69928 | 1 | i recently heard that there was a conference on Birch and Swinnerton dyer conjecture Held at Cambridge on May 4 until May 6,
the main theme is "The conference marks the 50th anniversary of the Birch-Swinnerton-Dyer Conjecture and aims both to explain the state of our knowledge and to reflect on the modern approaches to... | https://mathoverflow.net/users/nan | Where Can i find the lecture Videos of BSD 2011 | it seems that the lectures were not recorded (there are only photos on the conference website);
so there is no point in searching for the videos.
However, there are a few video excerpts available from Professor Stein's channel..small consolation.
(1) [bsd1](http://www.youtube.com/watch?v=Vo9yWAhBxAs)
(2) [bsd2... | 1 | https://mathoverflow.net/users/11786 | 69942 | 42,888 |
https://mathoverflow.net/questions/69946 | 9 | I was recently told that the following (due to M. Viale) is a nice theorem:
>
> Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper forcing and $V[G] \vDash \mathrm{MM}$. Then $L(P(\omega\_1))^V$ is elementarily equivalent to $L(P(\omega\_1))^{V[G]}$.
>... | https://mathoverflow.net/users/7521 | Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice? | By your tags you've asked for a soft answer, and so let me
try to provide one.
The theorem is indeed very nice and engages with and
reinforces a number of philosophical views in set theory.
First, there is the idea that large cardinal axioms are
leading us towards the final, true set-theory, and so
set-theorists ar... | 15 | https://mathoverflow.net/users/1946 | 69972 | 42,899 |
https://mathoverflow.net/questions/59837 | 7 | Let $G'$ be a graph obtained from $G$ after contracting each edge with probability $p$. Let $n = |V(G)|, e = |E(G)|$.
I would like to compute (or at least obtain a lower bound) for $E[|V(G')|]$ in terms of some known graph invariants (number of edges, degree sequence, connectivity,..)
I am sure I am not the first o... | https://mathoverflow.net/users/1737 | Randomly contracting edges of a graph - expected number of vertices? | The expected number of vertices of $G'$ is given by a sum (over all sets of edges) that bears a certain resemblance to the Tutte polynomial $T(x,y)$ of (the graphical matroid associated to) $G$, as defined, for example, at <http://en.wikipedia.org/wiki/Matroid#Tutte_polynomial>. A quick calculation (maybe too quick ---... | 5 | https://mathoverflow.net/users/6794 | 69973 | 42,900 |
https://mathoverflow.net/questions/69985 | 1 | Good morning,
I have just started reading Riemann surfaces. I would like to ask a question, maybe it is naive.
Let $X$ be a Riemann surface and $\phi\in\mathcal{O}\_{a,X}$ a holomorphic function germ at $a$ of $X.$ Let $u : [0,1]\to X$ be a curve, i.e a continuous mapping. Does it exist always an analytic continua... | https://mathoverflow.net/users/11376 | existence of analytic continuation | No, e.g. you may run into a singularity. For example, take $X = {\mathbb C}$, $u(t) = t$, $a=0$ and $\phi(z) = \frac{1}{1-2z}$ in a neighbourhood of 0. The pole at $t = 1/2$ stops the analytic continuation along the curve.
| 6 | https://mathoverflow.net/users/13650 | 69987 | 42,906 |
https://mathoverflow.net/questions/69988 | 7 | Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. Analogues have been proven for zeta function associated finite fields unconditionally.
The Riemann zeros with imaginary ... | https://mathoverflow.net/users/10400 | Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces? | You can look at the following survey by Peter Sarnak:
<http://www.math.princeton.edu/sarnak/Arithmetic%20Quantum%20Chaos.pdf>
Basically the prediction is that the eigenvalue distribution is Poisson for arithmetic surfaces and GOE for non-arithmetic surfaces. There are some partial results supporting Poisson in the ... | 16 | https://mathoverflow.net/users/16143 | 69990 | 42,908 |
https://mathoverflow.net/questions/69935 | 1 | Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a\_1,\dots,a\_n$. Then there is an isomorphism $\Phi: Hom\_R(A,B) \to Hom\_R(A,R)\otimes\_R B$ defined by $\Phi(\sigma)=\sum\_{i=1}^n \psi\_i\otimes \sigma\_i$, where $\sigma\_i=\sigma(a\_i)$ and $\psi\_i$ is ... | https://mathoverflow.net/users/4821 | $\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$ | Such an isomorphism is worthless if it is written down with a choice of a basis, because then naturality is unclear (which is, of course, very important if you need this isomorphism not just as an isolated relation). There is a homomorphism of $R$-modules
$\alpha : A^\* \otimes B \to \hom(A,B)$
defined by $\phi \ot... | 1 | https://mathoverflow.net/users/2841 | 69994 | 42,910 |
https://mathoverflow.net/questions/70002 | 13 | Any decent course on field theory will state that in characteristic $p$ an extension of fields $k\subset K$ canonically decomposes as the tower $k\subset K\_{sep}\subset K$ with
$K$ purely inseparable over its subfield $K\_{sep}$ of elements separable over $k$.
The alert student will then ask if the order can be re... | https://mathoverflow.net/users/450 | Is every algebraic extension of a field of absolute transcendence degree one a separable extension of a purely inseparable extension? | Yes. One may also replace $\mathbb{F}\_p$ by any perfect field $F$.
The reason is that any non-perfect algebraic extension $L$ of $k$ has a unique inseparable extension of degree $p$, i.e. $L^{1/p}$, the field obtained from $L$ by adjoining all $p$'th roots. This follows from the fact that $L^{1/p}$ is of degree at m... | 12 | https://mathoverflow.net/users/519 | 70003 | 42,917 |
https://mathoverflow.net/questions/70008 | 0 | Hi,
In his book *(Categories for the working mathematician)* MacLane speaks (on page 45) about the category of objects (of $\textbf{Ab}$) under $\mathbb{Z}$ which is the comma category $(\mathbb{Z}\downarrow \textbf{Ab})$, and says *"it is the category of abelian groups with a selected element"* (in analogy with $(\... | https://mathoverflow.net/users/2597 | The category of Abelian groups with selected elements | Pedro, sometimes we want to pick out an element or elements as an extra structure on abelian group. For example, we may be interested in a specified $\mathbb{Z}$-basis if the abelian group has one. Comma category constructions give a way of talking about such extra structure.
Of course, as you say there is already a... | 4 | https://mathoverflow.net/users/2926 | 70013 | 42,924 |
https://mathoverflow.net/questions/69933 | 3 | My question is if [this](https://mathoverflow.net/questions/68973/strict-transform-under-blow-up-along-nonsingular-subvariety "this") can be generalized as well:
Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}\_Z(X)\to X$ be the blow-up of a smooth algebraic variety $X$ along a irreducible generically s... | https://mathoverflow.net/users/15606 | Strict Transform under Blow-Up along singular Subscheme | I think the answer is, almost.
$E$ might have more than one component. Indeed, since you only assume that $Z$ is generically smooth, $E$ might have many different components. $E = \sum E\_i$.
You can always write $\beta^\* D = \widetilde{D} + \sum\_i \alpha\_i E\_i$, but you can't use the same $\alpha$ for all of ... | 5 | https://mathoverflow.net/users/3521 | 70016 | 42,927 |
https://mathoverflow.net/questions/70031 | 12 | Let us say that a mathematical structure of cardinality $\omega\_1$ is *Jonsson* whenever every one of its proper substructures is countable.
There are examples of Jonsson groups due to [Shelah](http://www.sciencedirect.com/science/article/pii/S0049237X08713466) or Obratzsov. I am almost sure that there is no Jonsson... | https://mathoverflow.net/users/15129 | Jonsson Boolean algebras? | Boolean algebras are never Jonsson.
Suppose that $\mathbb{B}$ is a Boolean algebra of size $\omega\_1$. Let $a$ be any element such that neither $a$ nor $\neg a$ is an atom. Note that every element $b\in\mathbb{B}$ is the join $b=(b\wedge a)\vee(b\wedge \neg a)$, and so there must be uncountably many elements either... | 28 | https://mathoverflow.net/users/1946 | 70033 | 42,935 |
https://mathoverflow.net/questions/69981 | 15 | Background: Let $X$ be a smooth complex projective algebraic variety, and let $V$ and $W$ be closed subvarieties. For simplicity, let's assume that $\dim V+\dim W=\dim X$.
Now Serre's famous Tor formula says that if $V\cap W$ has dimension zero, we have:
$$V\cdot W=\sum\_{Z\subset V\cap W}\sum\_{i=0}^\infty(-1)^i\o... | https://mathoverflow.net/users/35353 | Is there a Serre Tor formula for nonproper intersections? | There is no formula which looks only at the generic point(s) of $V \cap W$; you need to understand the entire sheaf $\mathcal{T}or\_j^{\mathcal{O}\_X}(\mathcal{O}\_V, \mathcal{O}\_W)$. It might be worth explaining the $K$-theory perspective on this.
---
Let $K\_0(X)$ be the Grothendieck group of coherent sheaves ... | 17 | https://mathoverflow.net/users/297 | 70037 | 42,937 |
https://mathoverflow.net/questions/69488 | 12 | By "surface bundle over a surface" I mean a compact, oriented 4-manifold $X$ which is the total space of an oriented fiber bundle $X\to B $ over an oriented 2-manifold $B$. Assume that the signature of the 4-manifold $X$ is non-trivial.
Conjecture 1: $X$ and $B$ can be given complex structures such that the map $X \... | https://mathoverflow.net/users/9617 | Are surface bundles over a surface with non-zero signature necessarily complex (or algebraic)? | I seem to have found an answer to my own question.
The first observation is that a holomorphic fibration is automatically algebraic. This is because $M\_g$ is quasi-projective and so the map $f:B\to M\_g$, which trivially extends to a map to $\overline{M}\_g$ is algebraic by GAGA (perhaps one should first pass to so... | 8 | https://mathoverflow.net/users/9617 | 70044 | 42,943 |
https://mathoverflow.net/questions/70004 | 1 | Let $f:\mathbb N\rightarrow\mathbb R$ be bounded. Let $\mu$ be a translation invariant finitely additive probability measure on $\mathbb N$.
**Question:** Are there any lower and upper bounds for $\int fd\mu$?
If $f$ is a characteristic function (or more generally if the image of $f$ is finite), one gets the lower... | https://mathoverflow.net/users/13809 | Bounds for the integral of a function with respect to an invariant measure | Define
$$
\overline\alpha(f)=\limsup\_{n-m\to\infty} \frac{1}{n-m}(f(m)+\ldots+f(n-1))
$$
and
$$
\underline\alpha(f)=\liminf\_{n-m\to\infty} \frac{1}{n-m}(f(m)+\ldots+f(n-1)).
$$
Claim $\max\int f\,d\mu=\overline\alpha(f)$ where the maximum is taken over translation-invariant finitely additive probability measures an... | 1 | https://mathoverflow.net/users/11054 | 70045 | 42,944 |
https://mathoverflow.net/questions/15316 | 45 |
>
> What is the length $f(n)$ of the shortest nontrivial group word $w\_n$ in $x\_1,\ldots,x\_n$ that collapses to $1$ when we substitute $x\_i=1$ for any $i$?
>
>
>
For example, $f(2)=4$, with the commutator $[x\_1,x\_2]=x\_1 x\_2 x\_1^{-1} x\_2^{-1}$ attaining the bound.
For any $m,n \ge 1$, the constructio... | https://mathoverflow.net/users/2757 | Collapsible group words | See the paper "Brunnian links" by Gartside and Greenwood, published in Fundamenta Mathematicae. Theorems 8 and 7 imply that iterated commutators are optimal and the sequence you suggest gives the minimal length.
| 19 | https://mathoverflow.net/users/1650 | 70048 | 42,945 |
https://mathoverflow.net/questions/70040 | 6 | A standard example of an ind-scheme over a field $\mathrm{k}$ which is not a
$\mathrm{k}$-scheme is $\mathrm{k}((\varepsilon))$.
My question is how to prove that rigorously? To put it more precisely,
let $$\mathrm{k}((\varepsilon)) = \{ a \in \prod\_{-\infty}^{\infty}\mathrm{k}:
a\_i =0, i \ll 0 \}$$
An ind-scheme i... | https://mathoverflow.net/users/16364 | How to show that an ind-scheme is not a scheme? | I think you can just consider the decreasing sequence of ideals of the increasing sequence of algebraic subsets $\epsilon^i k[[\epsilon ]]$ of $\prod\_{-\infty}^{\infty}k$. For each $i$, the ideal is $( a\_j | j\leq i-1 )$. The intersection of this sequence of ideals is $\{ 0 \}$. The corresponding algebraic subset of ... | 5 | https://mathoverflow.net/users/13265 | 70051 | 42,947 |
https://mathoverflow.net/questions/68625 | 6 | I am trying to compute the Hodge diamond of a Calabi-Yau fourfold which is a complete intersection inside of a projective bundle over a threefold base. I have computed the arithmetic genera $\chi\_1$ and $\chi\_2$ which give me two linear relations between the four independent Hodge numbers $h^{(1,1)}$, $h^{(1,2)}$, $h... | https://mathoverflow.net/users/15852 | Hodge diamond of a Calabi-Yau fourfold | I encountered a similar problem a few years ago, but then in dimension 3. In that case a master student wanted to calculate the hodge diamond of a threefold which was a hypersurface $W$ in a $P^2$-bundle over a del Pezzo surface. This threefold was birational to a singular hypersurface $Y$ in some weighted projective 4... | 10 | https://mathoverflow.net/users/8621 | 70052 | 42,948 |
https://mathoverflow.net/questions/70055 | 2 | I suspect this could be an easy one but I am not an expert in algebraic graph theory.
Let $Q(G)$ define the Laplacian matrix for a simple graph $G$. It is well known that n is an eigenvalue of $Q(K\_n)$ with multiplicity $n-1$. I was wondering if graphs $G$ of order $n$ such that $Q(G)$ has an eigenvalue of multiplic... | https://mathoverflow.net/users/1737 | Graphs of order n with a Laplacian eigenvalue of multiplicity n-1. | Such a graph has only two distinct eigenvalues, and graphs with few eigenvalues have been studied. See
<http://cage.ugent.be/geometry/Theses/30/evandam.pdf>
| 4 | https://mathoverflow.net/users/11142 | 70056 | 42,951 |
https://mathoverflow.net/questions/69050 | 4 | Hello,
I have an equation such as this. I do not have a Math background in my education, but I can read and understand Math pretty quickly. If I could be pointed to the right topics to solve this equation, it would be great. I want non-negative, integral solutions for the $a\_i$'s Also, s is a given integer.
$\dis... | https://mathoverflow.net/users/16074 | How to determine all non-negative solutions to a combinatorial diophantine equation? | I echo Richard Stanley in his pessimism for there being a nice formula or method for giving you the solutions. On the other hand, most of the solutions will have a\_i being 0 for most of the coefficients i. Here is another way to look at it, which might help you write a program for computer search for small n.
Your c... | 1 | https://mathoverflow.net/users/3402 | 70069 | 42,958 |
https://mathoverflow.net/questions/70071 | 3 | How does one prove that if $L/K$ is an extension of number fields with rings of integers $B/A$, then the module of Kahler differentials $\Omega^1\_{B/A}$ can be generated by one element as a $B$-module? When one proves that the annihilator of $\Omega^1\_{B/A}$ is the different of the extension, one localizes and comple... | https://mathoverflow.net/users/5498 | Module of Kahler differentials of rings of integers of number fields | You don't have to pick a $b\in B$ that works at all finite places. Indeed, such a $b$ might not exist. (I think there's an MO question about that.) But by the Chinese remainder theorem, a module over $B$ is cyclic if and only if it is so locally. So you can localize $B$, and then it's enough to check on the completion,... | 5 | https://mathoverflow.net/users/1114 | 70082 | 42,968 |
https://mathoverflow.net/questions/70090 | 2 | G is a finitely generated group and F is its subgroup.
Q: Under what known sufficient conditions can we guarantee that F is finitely generated?
(e.g. G is Abelian)
| https://mathoverflow.net/users/16323 | Finitely generated subgroup | A consequence of the Nielsen-Schreier theorem is the following: If a group generated by $n$ elements, then every subgroup of *finite* index $k$ is generated by $kn−k+1$ elements. See also [this](http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62075) aops discussion; there jmerry gives a direct algebraic proof.... | 9 | https://mathoverflow.net/users/2841 | 70092 | 42,972 |
https://mathoverflow.net/questions/69940 | 4 | **Question 1 (the weak and simple statement, which, I think, already is wrong):** Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical projection $V^{\otimes p}\to \mathrm{Sym}^p V$ from the $p$-th tensor power of $V$ to the $p$-th symmetric power of $V$... | https://mathoverflow.net/users/2530 | Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers? | $\newcommand{\SbV}{\mathrm{Sym}^2 V}$
$\newcommand{\ScV}{\mathrm{Sym}^3 V}$
$\newcommand{\quotA}{\left< v\otimes v\otimes v \mid v\in V\right>}$
$\newcommand{\quotB}{\left< vvv \mid v\in V\right>}$
I think I have solved this, with the help of mt and Tom Goodwillie.
Question 1 is wrong (and thus Question 2 is wrong ... | 1 | https://mathoverflow.net/users/2530 | 70102 | 42,978 |
https://mathoverflow.net/questions/70097 | 8 | We know that if $V=L$ holds, then $|\cal{P}(\omega)|=|\cal{P}(\omega)\cap \textrm{L}|=\aleph\_1$ whereas, in the presence of a measurable cardinal (in fact, even Ramsey) $|\cal{P}(\omega)\cap \textrm{L}|=\aleph\_0$. I remark that the cardinalities are of course computed in (the corresponding) $V$.
The first is just t... | https://mathoverflow.net/users/4826 | Large cardinals and constructible universe | Each of the following implies that (the true) $\omega\_1$ is inaccessible in $L$, and hence that there are only countably many constructible reals:
* The proper forcing axiom
* There is a Ramsey cardinal
* $0^\#$ exists
* All projective sets are Lebesgue measurable
* All $\Sigma^1\_3$-sets are Lebesgue measurable
... | 14 | https://mathoverflow.net/users/14915 | 70104 | 42,979 |
https://mathoverflow.net/questions/70106 | 2 | Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of several linear systems. For instance, if $h$ is a section of $\mathbb{P}^1$ and $f$ is a fibre, then I would be looking ... | https://mathoverflow.net/users/1887 | Global sections of a linear system | This question is actually a little bit vague. Anyway, I hope you can find the following answer useful.
One of the more general results about linear systems of curves on surfaces is the following theorem, proven by I. Reider in his paper [Vector bundles of rank $2$ and linear systems on algebraic surfaces (Annals of M... | 3 | https://mathoverflow.net/users/7460 | 70111 | 42,982 |
https://mathoverflow.net/questions/70093 | 4 | In Voinsin's book [1], Theorem 11.32 (page 280) says:
"If X is an algebraic variety, these subgroups of $Hdg^{2k}(X) coincide."
However, the proof did not show that the subgroup generated by
cycle classes (denoted by $A$) is containded in the subgroup generated by Chern classes of vector bundles (denoted by $B$) ... | https://mathoverflow.net/users/5093 | Why do the subgroups of $Hdg^{2k}(X)$ generated by the cycle classes and Chern classes of vector bundles coincide in algebraic variety $X$? | To expand slightly on Minhyong's comment, the key facts can be
found in Fulton's Intersection Theory. If you look at the comment following
corollary 18.3.2, you'll see an isomorphism (in slightly different notation)
$$ch:K^0(X)\otimes \mathbb{Q}\cong CH(X)\otimes\mathbb{Q}$$
where $X$ is a nonsingular variety, $K^0(X)$... | 10 | https://mathoverflow.net/users/4144 | 70113 | 42,983 |
https://mathoverflow.net/questions/69971 | 4 | I'm a CS student and I'm trying to learn RKHS theory to understand the passages made in [this paper](http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.74.6694) .
Among the bibliography I'm using there are ["On the mathematical fundamentals of learning"](http://www.ams.org/journals/bull/2002-39-01/S0273-0979-01-009... | https://mathoverflow.net/users/16348 | Doubts on Reproducing Kernel Hilbert Spaces and orthogonal decomposition | To answer your second question:
$H\_D = \operatorname{span}(k\_x : x \in D) $ is a finite dimensional (and hence closed) subspace of $H$. A function $f \in H$ is orthogonal to each $k\_x, x \in D$ precisely when $\langle f , k\_x \rangle = f(x) = 0$. In other words, $H\_D^\perp = ( f \in H : f(x\_i) = 0, x\_i \in D)$... | 3 | https://mathoverflow.net/users/16386 | 70122 | 42,988 |
https://mathoverflow.net/questions/70119 | 11 | I'm reading Kac's [Infinite Dimensional Lie Algebras](http://books.google.com/books/about/Infinite_dimensional_Lie_algebras.html?id=kuEjSb9teJwC). In [Chapter 4](http://books.google.com/books?id=kuEjSb9teJwC&pg=PA47&source=gbs_toc_r&cad=4#v=onepage&q&f=false), he classifies the affine root systems. [Bourbaki](http://bo... | https://mathoverflow.net/users/297 | Motivation behind Kac's notation for affine root systems | I think the notation might be explained by the explicit construction of the twisted affine Lie algebras as fixed points of automorphisms of the untwisted ones: the $r$ indicates the order of the chosen automorphism of the extended Dynkin diagram corresponding to $X$ and twised affine Lie algebra is a subalgebra of the ... | 13 | https://mathoverflow.net/users/519 | 70123 | 42,989 |
https://mathoverflow.net/questions/70120 | 5 | I have the conjecture that the volume of the intersection between an $n$-dim sphere (of radius $r$) and an ellipsoid (with one semi-axis larger than $r$) is maximized when the two are concentric, but still did not find a way to prove it. Any suggestion?
| https://mathoverflow.net/users/16385 | Maximize the intersection of a n-dimensional sphere and an ellipsoid. | From a [result of Zalgaller](http://mi.mathnet.ru/znsl686), this is true for any two centrally symmetric bodies.
([Here](http://www.mathnet.ru/php/seminars.phtml?option_lang=rus&presentid=169) is his lecture which inculdes this topic.)
Namely, assume that the center of first body is at $0$.
If $\vec r$ is the center ... | 11 | https://mathoverflow.net/users/1441 | 70131 | 42,992 |
https://mathoverflow.net/questions/69937 | 51 | I am doing my PhD in algebraic graph theory, for not much more reason than that was what was available. However, I love deep structure and theory in mathematics, and I do not particularly want to be a graph theorist for the rest of my life.
I have heard of mathematicians changing from more theoretical subjects to br... | https://mathoverflow.net/users/4078 | Changing field of study post-PhD | Speaking as someone whose thesis was also in algebraic graph theory but who has later gone on to do research in other areas, I would say that it is definitely possible to switch fields. The main skills you need are management skills: the ability to manage your own time so that you can spend some time learning a new fie... | 42 | https://mathoverflow.net/users/3106 | 70136 | 42,994 |
https://mathoverflow.net/questions/70061 | 3 | Let $X$ be an algebraic variety over $\mathbb{C}$ (or a normal complex space).
I found the word "equivariant resolution" in several papers on singularity theory or deformation theory. I think that it means the birational proper morphism of complex spaces $f: Y \rightarrow X $ where $Y$ is a complex manifold such that ... | https://mathoverflow.net/users/12390 | Reference on an equivariant resolution of singularities | Two other references:
Bierstone, Edward; Milman, Pierre D. (1997), "Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant.", Invent. Math. 128 (2): 207–302, doi:10.1007/s002220050141, MR1440306
Encinas, S.; Villamayor, O. (1998), "Good points and constructive resol... | 6 | https://mathoverflow.net/users/10696 | 70148 | 42,997 |
https://mathoverflow.net/questions/70035 | 6 | Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ and $b$ such that $|ax + by - z| < \epsilon$?
Note that if the restriction that $a$ and $b$ be coprime is lifted, the ... | https://mathoverflow.net/users/9021 | Searching for an inhomogeneous diophantine approximation algorithm | The problem isn't really about the existence of algorithms: The required coprime integers $a$ and $b$ can be found by a systematic search if they exist at all, assuming any reasonable interpretation of the word Given in the first sentence of the question.
It will simplify matters to divide the inequality in the first... | 4 | https://mathoverflow.net/users/5229 | 70150 | 42,999 |
https://mathoverflow.net/questions/70128 | 1 | The algebraic connectivity of a graph G is the second-smallest eigenvalue of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in t... | https://mathoverflow.net/users/10986 | A computation problem of algebraic connectivity of a tree | Firstly, your tree is just a Coxeter/Dynkin diagram of type $D$, so you may want to search MathSciNet with this in mind as your question might have been answered exactly at some point.
If you are happy enough with some bounds, it's easy to obtain $$\frac{4}{n(n-2)}\leq \lambda\_2(P)\leq \frac{12(n+2)}{n(n^2-1)},$$ s... | 3 | https://mathoverflow.net/users/14246 | 70153 | 43,001 |
https://mathoverflow.net/questions/70146 | 6 | Hello,
After reading the [previous post](https://mathoverflow.net/questions/17732/difference-between-measures-and-distributions), I still have some doubts. Let's consider everything on $R$ to avoid complications.
1. Can we say that any distribution $\mu\in\mathcal{D}'(R)$ of zero order is a signed radon measure?
2... | https://mathoverflow.net/users/36814 | Distributions and measures | To answer the first question: yes, at least locally. That is, given a distribution $u$ of order $0$, compactly supported for simplicity, the "order 0" condition asserts that $u$ factors through the space $C^o\_c(U)$ of continuous compactly supported functions \_with\_the\_corresponding\_topology\_. Then invoke the Ries... | 9 | https://mathoverflow.net/users/15629 | 70156 | 43,002 |
https://mathoverflow.net/questions/70126 | 13 | Disclaimer: This is a question I have not done any real research about. I asked it myself some 5 years ago, and back then I had no idea where to start. Now I have some texts on stable matchings lying around, but from a quick look they don't seem to answer this.
We have $n$ ladies $L\_1$, $L\_2$, ..., $L\_n$ and $n$ g... | https://mathoverflow.net/users/2530 | Gale-Shapley stable marriage theorem: can we entrust matchmaking to monkeys? | Regarding (2), the answer is still "no". The following counter-example is from:
>
> Tamura, Akihisa [Transformation from
> arbitrary matchings to stable
> matchings](http://www.ams.org/mathscinet-getitem?mr=1207740) J. Combin. Theory Ser. A
> 62 (1993), no. 2, 310–323
>
>
>
Consider $n$ men and $n$ women. W... | 11 | https://mathoverflow.net/users/297 | 70159 | 43,004 |
https://mathoverflow.net/questions/70154 | 10 | Hi all. Is there any explicit matrix expression for a general element of the special orthogonal group $SO(3)$? I have been searching texts and net both, but could not find it. Kindly provide any references.
| https://mathoverflow.net/users/16391 | Matrix expression for elements of $SO(3)$ | Here is the standard [quaternion](http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation) answer: Given $(a,b,c,d)$ such that $a^2+b^2+c^2+d^2=1$, the matrix
$$\begin{pmatrix}
a^2+b^2-c^2-d^2&2bc-2ad &2bd+2ac \\
2bc+2ad &a^2-b^2+c^2-d^2&2cd-2ab \\
2bd-2ac &2cd+2ab &a^2-b^2-c^2+d^2\\
\end{pmatrix}$$
is a rotation... | 23 | https://mathoverflow.net/users/297 | 70160 | 43,005 |
https://mathoverflow.net/questions/70140 | 10 | Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic functions as their Fourier coefficients.
More to the point, my question is, apart from the relations you obtain between $\si... | https://mathoverflow.net/users/16389 | Applications of full integral weight modular forms in elementary number theory | Perhaps this is "out of bounds" given the phrasing of the question, but those Eisenstein series you mention don't just have divisor sums as coefficients - the constant term is a special value of the Riemann zeta function.
This implies all sorts of neat stuff. The various relations between divisor sums that you mentio... | 10 | https://mathoverflow.net/users/12107 | 70180 | 43,016 |
https://mathoverflow.net/questions/70162 | 3 | According to Wikipedia, a Lawvere theory consists of a small category $L$ with (strictly associative) finite products and a strict identity-on-objects functor $I:\aleph\_0^\text{op}\rightarrow L$ preserving finite products.
Why a Lawvere theory have n-products for any n finite?
For example, why isn't a Lawvere theory... | https://mathoverflow.net/users/16393 | Why Lawvere theories have finite products? and more | I think you must have misunderstood how Lawvere theories are supposed to work. Let me try to motivate them from an algebraist point of view, and answer your question in passing.
An algebraic structure such as a group or a ring is usually described in terms of operations and equations they satisfy. But why do we choos... | 24 | https://mathoverflow.net/users/1176 | 70182 | 43,017 |
https://mathoverflow.net/questions/70171 | 2 | Considering in the complex fields. Let $P$ be a nonsingular matrix, $P^\* $ be its conjugate transpose, is there a relation between $P^\*|D|P$ and $|P^\*DP|$, where $D$ is a diagonal matrix? In particular, is it true
$P^\* |D|P \ge |P^\*DP|$ in the sense of Lowner order, or is there an order for eigenvalues?
Here ... | https://mathoverflow.net/users/6858 | Is there a relation between $P^*|D|P$ and $|P^*DP|$? | **Update:** In the edited question, the OP asks whether $\lambda\_i^\downarrow(P^\*|D|P) \ge \lambda\_i^\downarrow(|P^\*DP|)$ (or even the reverse direction). Such a relations do not hold either. Take for e.g., $P=\begin{bmatrix} 2 & 1 \\\\ 2 & 2\end{bmatrix}$, and use the same $D$ as below.
Then, we have $\lambda^\... | 7 | https://mathoverflow.net/users/8430 | 70184 | 43,019 |
https://mathoverflow.net/questions/69371 | 14 | Let $(M,g)$ be a Riemannian manifold of dimension $n>2$. Thanks to the late T.Branson we have the following definition of the so-called $Q$-curvature:
$Q= \Delta R + \frac{n^3-4n^2+16n-16}{4(n-1)(n-2)^2} R^2 - \frac{8(n-1)}{(n-2)^2}|Ric|^2.$
Here $\Delta = -div\nabla$, $R$ is the scalar curvature, and $|Ric|$ is th... | https://mathoverflow.net/users/15856 | Geometric Interpretation of $Q$-curvature | I think this is more like a remark than an answer. I gave seminar on Q-curvature (more precisely, Q-curvature flow) twice. In both seminars, I was asked, "What is the geometric meaning of Q-curvture? For example, if Q-curvature is zero, what can we conclude about the manifold $M$?" I was surprised that same question ha... | 5 | https://mathoverflow.net/users/14579 | 70188 | 43,021 |
https://mathoverflow.net/questions/69664 | 12 | It is known a version of Adjoint Functor Theorem for locally presentable categories, which says that a functor between such categories has a left adjoint iff it is continous (i.e. preserves all limits) and accessible (preserves $\lambda$-filtered colimits for some cardinal $\lambda$) (see for this Theorem 1.66 of J. Ad... | https://mathoverflow.net/users/15541 | Example of a continous functor between locally presentable categories which has no left adjoint | The following example was coincidentally mentioned by André Joyal on the categories mailing list today; he attributed it to Mac Lane. For every infinite cardinal number k, let $G\_k$ be a simple group of cardinality k. Define the functor ML: Group → Set to be the product of all the representable functors $\mathrm{Hom}(... | 16 | https://mathoverflow.net/users/49 | 70195 | 43,026 |
https://mathoverflow.net/questions/70198 | 7 | For the periodic Lorenz gas Sinai showed that rescaling the trajectory of the tracer particle yields Brownian motion in the limit. Does there exist a similar result for the random Lorenz gas? If not, do people believe that there is such a limit?
By the random Lorenz gas I mean: take circular scatterers distributed un... | https://mathoverflow.net/users/7949 | Does the random Lorenz gas have a non-trivial diffusion coefficient? | I don't think such a theorem has been proved for the random Lorentz gas. First I want to point out that Sinai proved those scaling limit results for the case of (2D) periodic Lorentz gas with *finite horizon* (finite maximum free path for the particle). The case of periodic Lorentz gas with unbounded horizon was studie... | 5 | https://mathoverflow.net/users/2384 | 70202 | 43,030 |
https://mathoverflow.net/questions/57773 | 8 | Let $n \in \mathbf{Z}\_{\geq 1}$. Is there an algorithm which, given a simplex $\Delta \subset \mathbf{R}^n$ specified as the convex hull of $v\_0,\dots,v\_n \in \mathbf{Z}^n$, computes the set $\Delta \cap \mathbf{Z}^n$ in time $O(\delta^t\mathrm{vol}(\Delta))$ for some explicit $t \in \mathbf{R}$, where $\delta=\max\... | https://mathoverflow.net/users/4433 | Listing lattice points in a simplex | Hermite normal form (HNF) should work much as you suggest.
Translate $v\_0$ to 0. Let $L = \bigoplus\_{i=1}^n {\bf Z} v\_i$. HNF gives an explicit decomposition of $G := {\bf Z}^n / L$ as a direct sum of cyclic groups. Both the computation and the resulting generators take time polynomial in the input size. This lets... | 5 | https://mathoverflow.net/users/14830 | 70204 | 43,031 |
https://mathoverflow.net/questions/70133 | 3 | Let $X/\mathbb{C}$ be a smooth projective curve of genus $g>0$ (here $\mathbb{C}$ is any algebraically closed field, say of characteristic $0$). Let $S$ be a finite set of closed points of X and let $\operatorname{Pic}^{level,S}(X)$ be the (commutative group) scheme (of infinite type) parametrizing line bundles on $X$ ... | https://mathoverflow.net/users/15630 | Global functions on generalized Jacobians | The assumption that $g>0$ is justified by the fact that the answer is clearly no in genus zero: in fact, if $C=\mathbb{P}^1$ and $S=\{\infty\}$, then the connected component of $\mathrm{Pic}^{\mathrm{level},\infty}(\mathbb{P}^1)$ is the affine group scheme $U$ with $U(\mathbb{C})=(1+u\,\mathbb{C}[[u]],\times)$ (where $... | 3 | https://mathoverflow.net/users/7666 | 70207 | 43,033 |
https://mathoverflow.net/questions/70211 | 7 | Given posets $P,Q\in M$, I would like to know under what circumstances there are mutually generic filters $G\subseteq P$ and $H\subseteq Q$ (generic over $M$). Also, what are the characterizations of mutual genericity? And finally, what can we say about the relation between $M[G]$ and $M[H]$ in that case?
I have a sl... | https://mathoverflow.net/users/4826 | Mutually generics | One of the basic facts of product forcing is the following,
which appears in any of the standard accounts of forcing:
**Theorem.** If $M$ is a model of ZFC and
$\mathbb{P},\mathbb{Q}$ are forcing notions in $M$, with
$M$-generic filters $G\subset\mathbb{P}$ and
$H\subset\mathbb{Q}$, then the following are equivalent:... | 13 | https://mathoverflow.net/users/1946 | 70212 | 43,034 |
https://mathoverflow.net/questions/70130 | 6 | My question pertains to exercise II-16 in Eisenbud and Harris' "The geometry of Schemes". For an algebraically closed field $K$ the question is as follows:
>
> Consider zero-dimensional subschemes $\Gamma \subset \mathbb{A}\_K^4$ of degree 21 such that $$V(m^3)\subset\Gamma \subset V(m^4)$$
> where $m$ is the maxi... | https://mathoverflow.net/users/16082 | Limits of reduced schemes question from Eisenbud and Harris | Such a subscheme is given by an ideal $I$ such that $m^4 \subset I \subset m^3$. In fact, any vector subspace in $m^3$ containing $m^4$ is an ideal (this is a simple exercise). Since $\dim O/m^3 = 15$ and $\dim O/m^4 = 35$, and we are interested in subspaces $I \subset O$ such that $\dim O/I = 21$, that is in subspaces... | 8 | https://mathoverflow.net/users/4428 | 70221 | 43,039 |
https://mathoverflow.net/questions/70142 | 1 | The de-Rham complex in one dimension describes phenomena that can be described in terms of ordinary differential equations. The de-Rham complex in three dimensions can be used to describe classical results in vector analysis.
Whereas the cochain morphism from the de Rham complex to complexes build out of scalar and v... | https://mathoverflow.net/users/2082 | Interpretation of the two-dimensional de-Rham complex | After writing this, I noticed that Donu Arapura made essentially the same point in a comment. But perhaps my added detail will be of use.
There are two missing ingredients in what you wrote.
First, when you identify 1-forms with vector fields, you have implicitly chosen a bundle isomorphism $T\mathbb{R}^n \cong T^... | 4 | https://mathoverflow.net/users/4362 | 70223 | 43,040 |
https://mathoverflow.net/questions/70127 | 2 | Does anybody know any classification of stable singularities of smooth map $f:\mathbb R^3\to \mathbb R^4$?
It is clear that there are singularities which look like intersection of 2 (or 3 or 4) hyperplanes in $\mathbb R^4$. But there are other ones.
Another type of stable singularity can be produced the following w... | https://mathoverflow.net/users/4298 | Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$ | The codimension of $\Sigma^2$-points in the source for a codimension $k$ map is $2(k+2)$, in this case that's $6>3$, so you only get combinations of Morin and regular points. From the Morins, the cusp has codimension $2(k+1)=4$, so you have no cusps either and need to keep track only of the regular points (codimension ... | 2 | https://mathoverflow.net/users/2368 | 70224 | 43,041 |
https://mathoverflow.net/questions/70225 | 2 | Let $G$ be a planar triangulation on $3m$ edges and $m+2$ vertices. Let $A$ be the binary matrix obtained from the incidence matrix of $G$ by deleting a row (equivalently we require the rows of $A$ to form a basis of the cocycle space of $G$ over $GF(2)$).
>
> My question is: What additional
> criteria must be ass... | https://mathoverflow.net/users/16414 | Planar Graphs and Skew Binary Spaces | What follows is all quite standard, iirc it's in Aigner's book "Combinatorics" but
my copy's out on loan....
Let $B$ be the vertex-edge incidence matrix of the graph.
(Nothing's gained by deleting a row.) We want a subspace of
codimension two in $GF(2)^{m+2}$ containing no column of $B$.
If $a$ and $b$ are distinc... | 2 | https://mathoverflow.net/users/1266 | 70238 | 43,048 |
https://mathoverflow.net/questions/70241 | 10 | Something that's always bothered me is that the word "transversal" is very commonly used as an adjective, but my understanding is that "transverse" is the correct adjective, and that "transversal" is a noun which means "an object which is transverse [to a given object]." So for example you would say "transverse interse... | https://mathoverflow.net/users/9417 | Terminology question: "Transverse" v. "Transversal" | "Transversal" is a good old geometry word, a noun, as you say. It goes way back to long before anybody was thinking of transversality in the modern sense.
It grates on me to hear it used as an adjective, and this owes something to the fact that in my impressionable youth I saw one of the chapter-heading quotations in... | 19 | https://mathoverflow.net/users/6666 | 70243 | 43,050 |
https://mathoverflow.net/questions/70226 | 4 | Some geometry problems ( like [this](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=38178) and [this](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=17) ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularl... | https://mathoverflow.net/users/13093 | How to solve geometry problems using involutions | I printed out your two link problems and looked at the language. It becomes clear that the word "involution" is used by "grobber" as any instance of a projective transformation that returns to the identity when repeated. This would include reflection in a fixed line, inversion across a fixed circle, rotation of 180 deg... | 3 | https://mathoverflow.net/users/3324 | 70258 | 43,058 |
https://mathoverflow.net/questions/70254 | 3 | Let $M$ be a ALE $n$-manifold. Then it is known a folklore result that
the $L^2$ Hodge cohomology is given by:
* $L^2\mathcal H^k=H^k(M,\partial M)$ if $k < n$,
* $L^2\mathcal H^{n/2}=Im(H^{n/2}(M,\partial M)\to H^{n/2}(M))$, and
* $L^2\mathcal H^k=H^k(M)$ if $k>n$.
A proof can be found in "HODGE COHOMOLOGY OF
GRAV... | https://mathoverflow.net/users/16423 | $L^2$ Hodge cohomology of ALE manifolds | I believe you can find this in the papers of Lockhart and McOwen. Specifically, check
Lockhart, Robert B.; McOwen, Robert C. Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409--447. MR0837256
[0837256](http://www.ams.org/mathscinet-getitem?mr=083725... | 4 | https://mathoverflow.net/users/6871 | 70261 | 43,061 |
https://mathoverflow.net/questions/70248 | 22 | Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in [A Survey of some Recent Developments in Differential Topology](http://www.ams.org/journals/bull/1963-69-02/S0002-9904-1963-10901-5/S0002-9904-1963-10901-5.pdf):
>
> Let f be a $C^\infty$ function on a closed manifold with no critica... | https://mathoverflow.net/users/2051 | Searching for an unabridged proof of "The Basic Theorem of Morse Theory" | R.Palais, [*Morse theory on Hilbert manifolds*](http://www.math.northwestern.edu/%7Egetzler/Morse/palais.pdf) (main Theorem of §12). As you will see, in the infinite dimensional setting the construction looses nothing in clearness.
| 17 | https://mathoverflow.net/users/6101 | 70265 | 43,062 |
https://mathoverflow.net/questions/70143 | 5 | Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines?
Everything I find in the literature is for blow-ups along smooth invariant centers.
Thanks!
| https://mathoverflow.net/users/1724 | What is the fan of the toric blow-up of $\mathbb{P}^3$ along the union of two intersecting lines? | The question is local near the intersection of the two lines, so the more basic question is to work this out for $\mathbb{A}^3$.
So we want to blow up $k[x,y,z]$ at $\langle z, xy \rangle$. There are two maximal charts:
$$\mathrm{Spec} \ k[x,y,z, (xy)/z] \ \mbox{and} \ \mathrm{Spec} \ k[x,y,z,z/(xy)].$$
Note tha... | 9 | https://mathoverflow.net/users/297 | 70269 | 43,063 |
https://mathoverflow.net/questions/70264 | 5 | Let $X$ be a finite-dimensional, locally compact topological space, and consider the dualizing complex $K\_X \in \mathbf{D}^b(X,k)$ (bounded derived category of $k$-sheaves, where $k$ is a noetherian ring). We can define the dualizing functor
$$C \mapsto D(C) = \mathbf{R}\mathcal{H}om(C, K\_X),$$
(derived internal hom)... | https://mathoverflow.net/users/344 | In what generality is the Verdier biduality map an isomorphism? | For an analytic space, you can find this on page 118 of Verdier's article "Classe d'homologie
d'un cycle" in Asterisque 36-37. And yes this is on an appropriate constructible derived category with $\mathbb{Z}$-coefficients. I seem to recall that Borel, in his book on intersection cohomology, also discusses this for pse... | 4 | https://mathoverflow.net/users/4144 | 70271 | 43,064 |
https://mathoverflow.net/questions/30201 | 6 | Let $G$ be an unramified reductive group over $Q\_p$. I want to prove that the group $G(Q\_p)$ has a supercuspidal representation (complex coefficients).
I have been looking in many parts of the literature, and it seems that many people are convinced that it is true; however up to now I never saw it stated explicitly... | https://mathoverflow.net/users/4398 | Existence of supercuspidal representations | Various special constructions make the point for particular groups, which tends to convince one about the general case. E.g., for $G=SL(2,\mathbb Q)$, a well-known (Shalika, Jacquet, c. 1970) idea is that inducing "supercuspidals" from $K=SL(2,\mathbb Z\_p)$ to $G$ is a finite direct sum of supercuspidals. The "level z... | 2 | https://mathoverflow.net/users/15629 | 70276 | 43,067 |
https://mathoverflow.net/questions/67858 | 11 | The following question was noted by Jan Pachl in connection with the
study of Arens products and he has not received a satisfactory
answer from the various experts he has asked. Let $X$ and $Y$ be
compact Hausdorff spaces and let $F$ be a continuous mapping from $X$
onto $Y$. Let
$A\subseteq Y$ and suppose that $F^{-1}... | https://mathoverflow.net/users/13878 | Images of Borel subsets of non-metric compact spaces | It is true that if $f:K \to L$ is a continuous mapping from a compact space $K$ onto $L$ and $A \subseteq L$ has Borel preimage in $K$ then $A$ is Borel in $L$. Jan Pachl points out that this is a very special case of a general theorem (Theorem 10) by
P. Holický and J. Spurný in "Perfect images of absolute Souslin and ... | 6 | https://mathoverflow.net/users/13878 | 70277 | 43,068 |
https://mathoverflow.net/questions/70208 | 0 | $p + p' = m$
$q - q' = n$
$pp' = qq'$
$(m^{2} + n^{2})\equiv1\pmod 4$ and $n^{2}\equiv0\pmod 4$.
Only $m,n$ are known in the above. Are there any known techniques to guess the values of $p$ and $q$ efficiently?
| https://mathoverflow.net/users/16007 | System of Diophantine equations | Since finding all possible expressions $$ M^2 + N^2 = m^2 + n^2$$ will in fact give you a complete factorization of $m^2 + n^2,$ all you need to know is how to create all $(M,N)$ pairs from a complete prime factorization of $m^2 + n^2 = p\_1 p\_2 p\_3 \ldots p\_r q\_1^2 q\_2^2 \ldots q\_s^2,$ where the $p\_i \equiv 1 \... | 5 | https://mathoverflow.net/users/3324 | 70286 | 43,072 |
https://mathoverflow.net/questions/70279 | 0 | Let $(M,g)$ be a Riemannian manifold and suppose that the Weyl tensor of $g$ vanishes at a point $p \in M$. Can one estimate the size of the largest geodesic ball around $p$ that we can make $g$ flat on through conformal deformation in terms of geometric data, i.e., $g$ and the usual curvature invariants? Is this just ... | https://mathoverflow.net/users/15856 | Estimate on the size of flat balls where the Weyl tensor vanishes | First, presumably you want to assume that the Weyl tensor vanishes not just at one point but everywhere, since at any point where the Weyl tensor does not vanish, there will be no conformal factor of the metric that is flat.
Second, I doubt what you want can be characterized using local pointwise invariants such as c... | 1 | https://mathoverflow.net/users/613 | 70287 | 43,073 |
https://mathoverflow.net/questions/69449 | 11 | When adding to the rational the $p$-torsion points $E[p]$ of an elliptic curve we obtain an extension containing the $p$-th roots of the unity, and whose Galois group can be embedded in $GL(2, \mathbb{F}\_p)$. To what extent are such extensions coming from elliptic curves?
I mean, assume $K/\mathbb{Q}$ to be an exten... | https://mathoverflow.net/users/3680 | Extensions obtained adding torsion points of an elliptic curve | This question is discussed very carefully in Section 3 of the paper *Mod $p$ representations
on elliptic curves*, by Frank Calegari (available [here](http://www.math.northwestern.edu/~fcale)).
In particular, after noting that the answer is positive when $p \leq 5$ (as was already
observed in the comments above), he ... | 14 | https://mathoverflow.net/users/2874 | 70288 | 43,074 |
https://mathoverflow.net/questions/70263 | 4 | I have two vectors: $\vec{a}$ and $\vec{t}$: $a\_k$ is the sampling value taken at $t\_k$. I need to do DFT, but the sampling period is irregular. I've learned about Frames but unsure how to use the Duffin–Schaeffer's suggestion for a Frame-function: $\sqrt{\frac{t\_{n+1}-t\_{n-1}}{2T}}sinc(\frac{\pi}{T}[t-t\_n])$.
D... | https://mathoverflow.net/users/16428 | How to do DFT for irregular sampling period ? | There have been a number of publications about "Unequally Spaced FFTs" in the numerical analysis literature. These typically involve an automated sort of interpolation (usually Gaussian) to an equally spaced grid followed by an FFT. The methods come with error bounds specified. You could search under the names Rokhlin,... | 4 | https://mathoverflow.net/users/8955 | 70290 | 43,075 |
https://mathoverflow.net/questions/70262 | 13 | Consider a finite-dimensional $\mathbf{Q}\_p$-vector space $V$ and a continuous representation $\rho : G\_{\mathbf{Q}\_p} \to \mathrm{GL}(V)$. Fontaine introduced various $\mathbf{Q}\_p$-algebras with $G\_{\mathbf{Q}\_p}$-actions, notated $B\_{\bullet}$ where $\bullet \in \left\{\mathrm{crys}, \mathrm{st}, \mathrm{dR}\... | https://mathoverflow.net/users/1464 | Is there a "trianguline period ring", or is one expected? | The category of trianguline representation is stable under all the usual representation-theoretic operations (subs, quotients, $\oplus$, $\otimes$), so by some general tannakian formalism, there does exist a ring $B\_{tri}$. The rough idea is to look at $Q\_p^{alg} \otimes B\_{st} \langle \langle \log(t) \rangle \rangl... | 14 | https://mathoverflow.net/users/5743 | 70299 | 43,079 |
https://mathoverflow.net/questions/70294 | 1 | In Voisin's book "Hodge theory and complex algebraic geometry I",
the proof of proposition 12.7 (page 296) says that
if $X$ is projective, then every divisor $Z$ homologous to $0$ can be written as a sum of divisors with multiplicity $1$.
Why is it true?
| https://mathoverflow.net/users/5093 | Why does the divisor $Z$ homologous to $0$ in projective mainfold satisfy that every irreducible hypersurface appears in $Z$ with multiplicity $1$? | I guess Voisin means that if $X$ is projective then every divisor $Z$ homologous to $0$ is *linearly equivalent* to a divisor $Z'$ which is a sum of divisors with multiplicity $1$.
In fact, the element $\alpha\_Z \in \textrm{Pic}^0(X)$ she wants to define only depends on the linear equivalence class of $Z$.
We star... | 4 | https://mathoverflow.net/users/7460 | 70302 | 43,081 |
https://mathoverflow.net/questions/69132 | 6 | I am looking for a reference (or an easy explanation) for the openness of the stable locus of a holomorphic family of (holomorphic) vector bundles on a compact Riemann surface parametrized by a (compact) complex manifold. For me, a holomorphic family of vector bundles on a compact Riemann surface $X$ parametrized by a ... | https://mathoverflow.net/users/509 | Reference for openness of stable locus of holomorphic family of vector bundles on a compact riemann surface. | I think this question is discussed in Kobayashi's "Differential geometry of complex vector bundles", at least the result follows implicitly.
One way to see is as follows: For simplicity, consider holomorphic rank 2 bundles $V$ of degree 0. They are not stable if there exists a holomorphic $f\colon L\to V$ of a holom... | 3 | https://mathoverflow.net/users/4572 | 70305 | 43,082 |
https://mathoverflow.net/questions/70210 | 8 | Consider a local non archimedean field $k$ and its ring of integer $o$. To what extent, do we know the complex irreducible representations of $GL(2,o)$? Is there a specific list giving them all in terms of induction from certain "simple" subgroups?
I have had read, that they have been classified according to their ch... | https://mathoverflow.net/users/10400 | To what extent do we know the representations of GL(2,Zp) | All irreducible representations of ${\rm GL}(2, {\mathcal O})$ have been constructed by Alexender Statinski :
Stasinski, Alexander (2009) The smooth representations of ${\rm GL}(2, {\mathcal O})$.
Reference in Math. Arxiv : <http://arxiv.org/abs/0807.4684>
But he does not follow the procedure that you propose at ... | 6 | https://mathoverflow.net/users/4767 | 70306 | 43,083 |
https://mathoverflow.net/questions/70300 | 1 | Hi
I'm not quite sure what title to give to this question or what tags to use, because this isn't really my area of expertise and I'm unfamiliar with the terminology. It is a problem that came up while trying to write a program to enumerate some graphs, however there is no graph theory left in this problem.
Let me ... | https://mathoverflow.net/users/15684 | Algorithm to determine a discrete function | I will assume that $C$ is finite for my answer.
Let $n,d:C\to\mathbb{N}$ such that $n\equiv\left.m/r\right.$ and $r\equiv\left.s/d\right.$. In other words, $\mathrm{im}(d)\subseteq\{1,2\}$. Let $C\_1:=d^{-1}(1)$ and $C\_2:=C\setminus C\_1 = d^{-1}(2)$. Now, we want to find a function $n$ such that
$R := \displaysty... | 2 | https://mathoverflow.net/users/9947 | 70310 | 43,085 |
https://mathoverflow.net/questions/70304 | 7 | This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $G\subset\mathbb{Z}^n$ be a subgroup, and $G\_+=G\cap\mathbb{Z}\_{\ge0}^n$ be a cone of elements in $G$ whose coordinates are all nonnegative. The semigroup $G\_+$ is finitely generated; it can be ... | https://mathoverflow.net/users/1306 | Positive cone of a subgroup of $\mathbb{Z}^n$ | No, there is no upper bound on the number of generators in terms of $n$.
Let $k$ be arbitrary positive integer.
Consider the subgroup
$$G=\{(x,y)\in\mathbb Z^2\mid x+y\equiv 0\pmod k\}.$$
Any set of generators of $G\_+$
contains all the elements $(x,y)$ such that $x,y\ge 0$ and $x+y=k$.
Therefore the number of gener... | 11 | https://mathoverflow.net/users/1441 | 70314 | 43,087 |
https://mathoverflow.net/questions/70311 | 4 | Are there some nontrivial group homomorphisms from $SL\_n(\mathbb{Z})$ to $GL\_{n-1}(\mathbb{Z})$ for $n\geq3$ except the determinant? This should be a natural question and any references are welcomed.
PS. A similar question has the answer 'NO' for a finite field $F\_p$ instead of $\mathbb{Z}$ as explained below.
| https://mathoverflow.net/users/1546 | Are there some nontrivial group homomorphisms from $SL_n(\mathbb{Z})$ to $GL_{n-1}(\mathbb{Z})$ for $n\geq3$? | Se [Does $SL\_3(R)$ embed in $SL\_2(R)$?](https://mathoverflow.net/questions/47407/does-sl-3r-embed-in-sl-2r/) for a related discussion.
That any homomorphism $\varphi\colon SL\_n(\mathbb{Z}) \to GL\_{n-1}(\mathbb{Z})$ is trivial can be seen as follows.
By Margulis' normal subgroup theorem, either the kernel of $... | 7 | https://mathoverflow.net/users/3380 | 70315 | 43,088 |
https://mathoverflow.net/questions/70272 | 4 | What is the official name of this problem? Martin Gardner gives introduction in his book "Math circus". The problem belongs to 1D random walk. What can be read to gain deep insight into this problem? Or other useful resources.
>
> We can complicate matters by allowing transition probabilities
> to vary from 1/2 a... | https://mathoverflow.net/users/16430 | Puzzle in Martin Gardner book | Sure, multiplication is commutative, but there is more to it than that. While being reasonably easy, this puzzle suggests variations in ways that the equation $x\cdot y = y\cdot x$ doesn't.
In his wonderful paper Games People Don't Play, <http://www.teorver.ru/newkatalog/1193689162.pdf>, Peter Winkler describes esse... | 1 | https://mathoverflow.net/users/14302 | 70317 | 43,090 |
https://mathoverflow.net/questions/70282 | 10 | Hello,
If we possess the eigendecomposition of a positive definite matrix: $X = U \Sigma U^T$, is there an efficient way to compute the eigendecomposition of $D X D$ where $D$ is a diagonal matrix?
| https://mathoverflow.net/users/16433 | Eigendecomposition after multiplying by diagonal matrix | Write $\Sigma$ as $T^2$, for positive definite $T$. Set $Y = U T$.
So the eigenvalues of $X$ are the squares of the [singular values](http://en.wikipedia.org/wiki/Singular_value_decomposition#Singular_values.2C_singular_vectors.2C_and_their_relation_to_the_SVD) of $Y$, and what you want to compute are the singular v... | 9 | https://mathoverflow.net/users/297 | 70321 | 43,091 |
https://mathoverflow.net/questions/70318 | 9 | I'm a string theorist and I have come across the following expression in a computation I'm doing (involving a sum over inequivalent Lens spaces):
$$\widehat{\zeta}(s)=\prod\_{\mathrm{primes}\ p\equiv 3\ (\mathrm{mod}\ 4)}(1-p^{-s})^{-1}.$$
I expected this to be a standard sort of construction, and it's clearly closely ... | https://mathoverflow.net/users/9963 | Euler product over primes congruent to 3 mod 4 | Your functional equation shows that $\widehat{\zeta}(s)^2$ has a meromorphic continuation to $\Re(s)>1/2$ with a simple pole at $s=1$. This also shows that $\widehat{\zeta}(s)$ does not have a meromorphic continuation to a neighborhood of $s=1$, instead it lives on a double cover of that neighborhood branched at $s=1$.... | 7 | https://mathoverflow.net/users/11919 | 70328 | 43,095 |
https://mathoverflow.net/questions/70335 | 2 |
>
> **Possible Duplicate:**
>
> [Is there an example of a scheme X whose reduction X\_red is affine but X is not affine?](https://mathoverflow.net/questions/95/is-there-an-example-ofa-scheme-x-whose-reduction-x-red-is-affine-but-x-is-not-af)
>
>
>
I got a question, which may be very easy, but I didn't figure... | https://mathoverflow.net/users/5482 | a question about affiness | If $X$ is noetherian the answer is *yes*.
Indeed, in this case $X$ is affine if and only if $X\_{\textrm{red}}$ is affine.
See [Hartshorne, Algebraic Geometry], Exercise 3.1 p. 216.
| 3 | https://mathoverflow.net/users/7460 | 70336 | 43,097 |
https://mathoverflow.net/questions/70337 | 6 | Good Morning,
I'm trying to prove that two different definitions of the Hirzebruch Surfaces coincide, and am having problems. Let $a \geq 0$. My first definition for the $a^{th}$ surface is
$X\_a= \mathbb{P}(\mathcal{O}(a) \oplus \mathcal{O}) \longrightarrow \mathbb{P}^1\_{\mathbb{C}}$
My second definition is as ... | https://mathoverflow.net/users/8867 | Hirzebruch Surfaces | The cone $D\_a$ has a singular point of type $\frac{1}{a}(1,1)$ at its vertex. Blowing up the vertex, the exceptional divisor is a curve $C \subset Y\_a$ isomorphic to $C\_a$ and whose self-intersection is $\deg \mathcal{O}\_{C\_a}(-1)=-a$.
Since $Y\_a$ is clearly a geometrically ruled surface over a rational curve ... | 7 | https://mathoverflow.net/users/7460 | 70340 | 43,099 |
https://mathoverflow.net/questions/70347 | 11 | While working on a project, I have run into a situation where I have integers x and n so that $x^n \equiv (x+1)^n \equiv (x+2)^n$ mod $p$ for a prime $p$. It seems to me that this an extremely restrictive condition, and I was wondering if there are any results about when (or if?) it can happen, but I couldn't figure ou... | https://mathoverflow.net/users/4535 | Successive nth powers mod p? | Write $a = (x+1)/x, b = (x+2)/x$, then your condition is equivalent to $a^n \equiv b^n \equiv 1 \mod p$ and $2a-b \equiv 1 \mod p$. Now, without loss of generality, you can assume $n|(p-1)$ (otherwise replace $n$ by the gcd of $n$ and $p-1$). Write $m = (p-1)/n$ and $a=u^m,b=v^m$. Finally, your conditions become $2u^m-... | 26 | https://mathoverflow.net/users/2290 | 70351 | 43,102 |
https://mathoverflow.net/questions/66641 | 20 | I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any reference that treat systematically the relation between such models of theories, where model means a presentation of theory?
... | https://mathoverflow.net/users/14969 | Relation between monads, operads and algebraic theories | All nice recommendations; some more
### Lawvere theories and monads
For the connection between monads and Lawvere theories, I've found this a really nice exposition of the $\mathbf{Set}$ case:
>
> Martin Hyland, John Power - *The category theoretic understanding of universal algebra: Lawvere theories and monads... | 10 | https://mathoverflow.net/users/4315 | 70352 | 43,103 |
https://mathoverflow.net/questions/70343 | 3 | I am a software engineer. I'm dealing with a data structure which represents a digraph of a very specific structure. I am wondering if this class of graph has been identified and studied as I need to do a fair bit of work with it and would love to not reinvent the wheel if not necessary.
The structure of the graph is... | https://mathoverflow.net/users/16452 | Graph Theory: question regarding a class of digraph | Qiaochu Yuan has already provided an answer in terms of functions, but if you prefer to think graph-theoretically these things are known as [directed pseudoforests](http://en.wikipedia.org/wiki/Pseudoforest).
| 7 | https://mathoverflow.net/users/440 | 70356 | 43,106 |
https://mathoverflow.net/questions/70242 | 0 | Given $\mathcal{O}=k[[u,v]]$ with maximal ideal $\mathfrak{m}$ and an $\mathcal{O}$-algebra $A$, free as an $\mathcal{O}$-module of rank $n^2$. $A$ is genertaed by two elements $x,y$ with $x^n=u$, $y^n=v$ and $xy=\xi\_n yx$, where $\xi\_n$ is an n-root of unity. (But i think this should be true in more general situatio... | https://mathoverflow.net/users/3233 | Generalized Picard group (reflexive fractional ideals, principal ideals) | Let $M$ be a coherent (left) $\mathcal{A}$-module which is Zariski locally isomorphic to $\mathcal{A}$. Then in particular, the stalk at the generic point of $U$ is isomorphic to the stalk of $\mathcal{A}$ at the generic point. Choose such an isomorphism. By adjunction, this is equivalent to an $\mathcal{A}$-module hom... | 2 | https://mathoverflow.net/users/13265 | 70362 | 43,109 |
https://mathoverflow.net/questions/70308 | 10 | Is the following statement proved?
For any positive integer $k$ there exists positive integer $n$ such that all sufficiently large integers may be represented as $p\_1^k+p\_2^k+\dots+p\_n^k$ for primes $p\_1,\dots,p\_n$.
[This Wiki article](https://en.wikipedia.org/wiki/Waring-Goldbach_problem) claims that some pro... | https://mathoverflow.net/users/4312 | Current status of Waring-Goldbach problem | This is a corrected version of my original response, incorporating a nice argument by Fedor Petrov.
Hua in his book (cf. review of MR0124306) proved that there are integers $s,K,N>0$ such that every $n>N$ with $n\equiv s\pmod{K}$ is a sum of $s$ $k$-th powers of primes. For any $t>0$ let $M(t)$ denote the set of resi... | 10 | https://mathoverflow.net/users/11919 | 70372 | 43,115 |
https://mathoverflow.net/questions/70176 | -1 | Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C\_c^\infty(R)$ (smooth functions with compact support)?
Following the suggestion by André Henriques, I will put the answer in th... | https://mathoverflow.net/users/36814 | Can singular measures be viewed as vanishing distributions? (Answer No!) | After reading comments by Andreas and Wong, I understand that the above statement is wrong. Professor Gerald Edgar gives the simplest example (at least for me :-) ): delta function is a singular measure, but for any test function $\psi$ not vanishing at zero $\psi(0)\ne 0$, we have that $\int \psi \delta(d x)=\psi(0)\n... | 0 | https://mathoverflow.net/users/36814 | 70378 | 43,117 |
https://mathoverflow.net/questions/70376 | 7 | I'm never sure about free groups whether a question is easy or not. It feels to me like this is impossible, but I couldn't come up with any argument.
If $F\_n$ is a free group on $n$ generators, could there exist a non-trivial $N\triangleleft F\_n$ such that $F\_n/N \cong F\_n$?
| https://mathoverflow.net/users/5756 | Can a finitely generated free group be isomorphic to a non-trivial quotient of itself? | No, the free groups are [Hopfian](http://en.wikipedia.org/wiki/Hopfian_group) because they are [residually finite](http://en.wikipedia.org/wiki/Residually_finite_group).
| 14 | https://mathoverflow.net/users/nan | 70380 | 43,118 |
https://mathoverflow.net/questions/70384 | 5 | Suppose that a 2-category $\mathcal{C}$ has strict pullbacks and one has maps $f:F\to C$, $g\_0,g\_1:G\to C$ and a natural transformation $\gamma:g\_1\implies g\_0$. Is there a good notion of a pullback transformation $f^\*\gamma$. If so, I would expect its codomain to be $$f^\*g\_1\times\_G f^\*g\_0\to f^\*g\_1\to F$$... | https://mathoverflow.net/users/16466 | Is it possible to pull back a natural transformation? | There is a good notion of a pullback transformation, but its domain and codomain aren't what you guessed; rather one asks for a map from $f^\*(g\_0) \to f^\*(g\_1)$ and a 2-cell filling the resulting triangle over G. The existence of such a pullback transformation is also not automatic from the existence of pullbacks, ... | 6 | https://mathoverflow.net/users/49 | 70395 | 43,125 |
https://mathoverflow.net/questions/70385 | 3 | Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu\_+$ and negative one $\mu\_-$ by [the Hahn decomposition theorem.](http://en.wikipedia.org/wiki/Hahn_decomposition_theorem)
My question is whether each real-valued Radon measure $\nu$ on $R$ can be decomposed by the pos... | https://mathoverflow.net/users/36814 | Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures? | I'd take the view that a real-valued Radon measure "really is" a continuous linear functional $\Lambda:C\_{00}(X,\mathbb R) \rightarrow \mathbb R$. Here $X$ is a locally compact space, and $C\_{00}(X,\mathbb R)$ is the collection of compactly supported continuous real valued functions on $X$. By "continuous", we mean t... | 5 | https://mathoverflow.net/users/406 | 70426 | 43,139 |
https://mathoverflow.net/questions/70425 | 1 | With parameters: srg(v(v-1)/6, 3(v-3)/2, (v-3)/2, 9)
Should be straightforward counting which alludes me...
Thanks!
Shay
| https://mathoverflow.net/users/16480 | Why is a block graph of a Steiner Triple System is a Strongly Regular Graph? | It's been a time since I learned something about STS, but let's have a try.
There are of course $\frac{v(v-1)}{6}$ vertices in your srg since there are $v(v-1)$ pairs when you have $v$ elements, and there are 6 pairs when you have 3 elements, so each triple is counted 6 times.
Each element is contained in $v-1$ pai... | 2 | https://mathoverflow.net/users/15684 | 70433 | 43,144 |
https://mathoverflow.net/questions/70231 | 9 | Let $R$ be a ring, $A$ a (not necessarily commutative) $R$-algebra and $M$ a (left) A-module which is free of finite rank as an $R$-module. If $a\in A$ then multiplication with $a$ on $M$ is an $R$-linear endomorphism of $M$ and as such it has a characteristic polynomial $\chi\_a$. I've learned from notes by Bart de Sm... | https://mathoverflow.net/users/2308 | Reference for a formula expressing the characteristic polynomial of a sum of endomorphisms | One reference is: S. A. Amitsur, On the Characteristic Polynomial of a Sum of Matrices,
Linear and Mult. Algebra 8 (1980), 177-182. (pp. 469-474 in Selected Papers of S. A. Amitsur,
Part 2, AMS 2001.)
| 7 | https://mathoverflow.net/users/9347 | 70439 | 43,149 |
https://mathoverflow.net/questions/70429 | 13 | For a $n$-dim smooth projective complex algebraic variety $X$, we can form the complex line bundle $\Omega^n$ of holomorphic $n$-form on $X$. Let $K\_X$ be the divisor class of $\Omega^n$, then $K\_X$ is called the canonical class of $X$.
**Question**: Is homology class of $K\_X$ in $H\_{2n-2}(X)$ a topological invar... | https://mathoverflow.net/users/15289 | Is canonical class a topological invariant? | This answer is about the case of complex surfaces $X$ and their diffeomorphisms (all my diffeos are assumed to be orientation-preserving!).
**(1) Examples of self-diffeomorphisms that reverse the sign of the canonical class.**
Take $X=\mathbb{C}P^1\times \mathbb{C}P^1$. Let $\tau$ be reflection in the equator of ... | 13 | https://mathoverflow.net/users/2356 | 70445 | 43,153 |
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