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https://mathoverflow.net/questions/77177 | 6 | Let A be a quasinilpotent operator on a Hilbert space and let $A^{\*}A$ have finite spectrum.
Does then follow, that A is nilpotent ?
| https://mathoverflow.net/users/17261 | quasinilpotence and finite spectrum | I found a counterexample :
Let $e\_{1},e\_{2},...$ be ON basis of the Hilbert space and define A by
$Ae\_{2n-1} = \sqrt{1-\frac{1}{n^{2}}} \ e\_{2n} \ + \ \frac{1}{n}
\ e\_{2n+1}$ , $\ \ $n=1,2,3,... ,
$Ae\_{2n} = 0$ , $\ \ $n=1,2,3,...
Then A is a partial isometry and therefore $A^{\*}A$ a projection.
Furt... | 2 | https://mathoverflow.net/users/17261 | 77384 | 46,762 |
https://mathoverflow.net/questions/77383 | 28 | I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is just that nobody thought enough about that, or maybe it is not a stupid question.
**Question:** Do there exist two Ba... | https://mathoverflow.net/users/13809 | A separable Banach space and a non-separable Banach space having the same dual space? | The duals of $C[0,1]$ and of $C[0,1]\oplus\_\infty c\_0(\Bbb{R})$ are isometrically isomorphic.
| 37 | https://mathoverflow.net/users/2554 | 77385 | 46,763 |
https://mathoverflow.net/questions/77343 | 5 | Marshall Hall, in his famous book *Theory of Groups*, does not always require a binary operation be "well-defined", i.e. an operation is a relation instead of a function (there might be more than one $c$ s.t. $ab=c$). For example, in the discussion of "quasi-group with the inverse property", he use the inverse property... | https://mathoverflow.net/users/3332 | Hall's treatment of algebraic operations | I am looking at page 7 of Hall's book and I do not see that he is using binary operation in
any non-standard fashion. He defines a "quasigroup with the inverse property" to be a set
with a binary product and unary inverse such that
$$
a^{-1}(ab) = b = (ba)a^{-1}.
$$
He then wants to show that this is a quasigroup, by ... | 6 | https://mathoverflow.net/users/1266 | 77387 | 46,765 |
https://mathoverflow.net/questions/77389 | 8 | This seems like it must have been addressed somewhere already, but I cannot find it in any standard series tables.
I have the equation:
$f(z) = \left(1 + \frac{1}{z}\right)^z$.
What is the general form for the $n$th term of the series? That is, if I have
$f(z) \sim \sum\_{n=0}^{\infty} \frac{c\_n}{z^n}$
near... | https://mathoverflow.net/users/18345 | What is the series expression for (1+1/x)^x about x = \infty? | Markus Brede proves the following formula in the paper ["On the convergence of the sequence defining Euler’s number"](http://www.springerlink.com/content/c1q27x0730v30hth/). Let $$\left(1+\frac{1}{z}\right)^z=\sum\_{n\geq 0} \frac{a\_n}{z^n}$$
then we have
$$a\_n=e\sum\_{v=0}^n \frac{S(n+v,v)}{(n+v)!}\sum\_{m=0}^{n-v}\... | 14 | https://mathoverflow.net/users/2384 | 77397 | 46,769 |
https://mathoverflow.net/questions/77400 | 10 | The Borel Conjecture asserts that homotopy equivalent aspherical closed manifolds are homeomorphic, which is still open in general.
But, for three-dimensional manifolds, this conjecture holds (I read this in [Bessieres-Besson-Boileau](http://www-fourier.ujf-grenoble.fr/~lbessier/english_principal.pdf)), whose proof dep... | https://mathoverflow.net/users/2879 | Topological rigidity of compact manifolds in dimension three | Yes.
When the manifolds are Haken this is a theorem of Waldhausen. See Ian Agol's answer [here](https://mathoverflow.net/questions/35680/complete-knot-invariant/35687#35687).
Since your manifolds are aspherical, they are irreducible by the Poincaré conjecture. Since they have boundary and are irreducible, they are ... | 12 | https://mathoverflow.net/users/1335 | 77404 | 46,772 |
https://mathoverflow.net/questions/77391 | 6 | What is the [reverse mathematics](http://en.wikipedia.org/wiki/Reverse_mathematics) strength of
"For all [Lipschitz](http://en.wikipedia.org/wiki/Lipschitz_continuity) functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ?
(defined using e... | https://mathoverflow.net/users/nan | reverse mathematics strength of "Lipschitz functions are somewhere differentiable" | (Update: I made my answer clearer and also fixed the references and added more.)
Your theorem should be true in every $\omega$-model of $\mathsf{RCA}\_0$ as follows. The following paper
* Brattka, Miller and Nies. *Randomness and Differentiability*. Submitted. ([preprint](http://dl.dropbox.com/u/370127/papers/rando... | 7 | https://mathoverflow.net/users/12978 | 77407 | 46,774 |
https://mathoverflow.net/questions/77409 | 4 | Let $k$ be a field and let $R=k[x\_1,...,x\_n]$ be a polynomial ring over $k$. A subset $\lbrace y\_1,...,y\_n\rbrace$ of $R$ is called a homogenous system of parameters (hsop) of $R$, if
* the $y\_i$ are homogenous polynomials
* $R$ is finitely generated as $k[y\_1,...,y\_n]$-module
I am looking for criteria, whe... | https://mathoverflow.net/users/10194 | Criteria for system of parameters in polynomial rings | Recall that the resultant $Res(g\_1,\cdots, g\_n)$ of $n$ (not necessarily homogenous) polynomials in $n-1$ variables is $0$ if and only if the $g\_i$s have a common root.
Now, the $y\_1,\cdots, y\_n$ form an sop iff $(0,\cdots, 0)$ is the only common root. But if they have another common root, then because of homog... | 3 | https://mathoverflow.net/users/2083 | 77417 | 46,778 |
https://mathoverflow.net/questions/77439 | 3 | What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference?
To go from a self-duality $F$ on a category to a groupoid, I guess I have to check for $f: A \to B$ that $F(f) \circ f$ and $f \circ F(f)$ are the identities ... | https://mathoverflow.net/users/18355 | A category being self-dual vs. it being a groupoid | The category $\mathbf{2}$ with two objects ($0$ and $1$, say) and just one map between them ($0 \to 1$) is self-dual and clearly not a groupoid. To see that it is self-dual, just consider the functor that exchanges $0$ and $1$. It is clearly its own quasi-inverse. If it were a groupoid, then the objects $0$ and $1$ wou... | 8 | https://mathoverflow.net/users/6348 | 77441 | 46,792 |
https://mathoverflow.net/questions/76591 | 4 | This question is related to the question: [Is there a $k$-structure for Hodge modules over a $k$-variety?](https://mathoverflow.net/questions/45948/is-there-a-k-structure-for-hodge-modules-over-a-k-variety).
Suppose $K$ is a subfield of $\mathbb{C}$ and $M$ is a holonomic $D$-module "of geometric origin" on a smooth ... | https://mathoverflow.net/users/15630 | Hodge theory and varieties defined over subfields of the complex numbers | Take a look at section 1 of
M. Saito, Arithmetic mixed sheaves, Inventiones 2001. Part of the data
for an object in his category $MHM(X/K)$ is a bifiltered $D$-module defined over $K$. So it seems that your question has a positive answer. But as always with this stuff, it is easy to overlook something. If it's really ... | 4 | https://mathoverflow.net/users/4144 | 77443 | 46,793 |
https://mathoverflow.net/questions/77447 | 1 | Dear community,
I have a problem which is very simple to state but seems to be hard to answer.
Statement of the problem
------------------------
Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, both of them square integrable over $T^n$ and $T^m$, respectively, where $T$ is some subset o... | https://mathoverflow.net/users/12366 | Can symmetrizing a contraction increase the speed of convergence? | Actually, it is even possible that $\widetilde{f \otimes\_1 g}\equiv0$ without $f \otimes\_1 g$ being zero. Just consider a domain $T$ consisting into two subdomains of unit volume, and functions $f$ and $g$ that are constant on each of these. Then it is the same as dealing with symmetric $2\times2$ matrices $F$ and $G... | 1 | https://mathoverflow.net/users/8799 | 77450 | 46,796 |
https://mathoverflow.net/questions/77418 | 13 | $\DeclareMathOperator\SO{SO}\DeclareMathOperator\so{\mathfrak{so}}$Consider the Lie algebra $\so(n)$ equipped with the metric $\langle e\_i \wedge e\_j, e\_k \wedge e\_l \rangle = \delta\_{i,k} \delta\_{j,l}$. Similarly equip the tangent space at other points of $\SO(n)$ by left translation. My question is, is the expo... | https://mathoverflow.net/users/4923 | What's the Lipschitz constant of the exponential map for $\mathrm{SO}(n,R)$? | This is also true for the Riemannian metric described in the original question. I'm sure there is an elementary argument to prove it - it should certainly follow from the formula for the differential of the matrix exponential for example but here is a quick argument which involves no computations at all (assuming you k... | 19 | https://mathoverflow.net/users/18050 | 77456 | 46,799 |
https://mathoverflow.net/questions/77449 | 5 | Characteristic polynomials of Hecke operators $T\_\ell$, with $\ell$ prime, acting on cusp forms $S\_k$ of level one and weight $k$ "appear to be" squarefree (even irreducible!).
This can be interpreted as follows: if $p$ is any prime, then the $p$-adic Galois representations $\rho\_i$, where $1\leq i\leq \dim(S\_k)$... | https://mathoverflow.net/users/4800 | Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$ | I imagine this is a weight issue, not a level issue. Let f and g be the two weight-2 newforms in $S\_2(\Gamma\_0(37))$. Then a "random" coefficient of f-g is going to have size about $p^{1/2}$, so there should be about $X^{1/2-\epsilon}$ primes $p$ less than $X$ such that $a\_p(f) = a\_p(g)$, just by chance. When the w... | 6 | https://mathoverflow.net/users/431 | 77457 | 46,800 |
https://mathoverflow.net/questions/77446 | 15 | Background
----------
In the book Problems in Modern Mathematics, S. Ulam asks the following question:
Suppose $A$ and $B$ are metric spaces, such that $A^2$ and $B^2$ equipped with the 2-metric $d((x\_1, y\_1),(x\_2, y\_2)) = \sqrt{d(x\_1, x\_2)^2 + d(y\_1, y\_2)^2}$ are isometric. Does it follow that $A$ and $B$ ... | https://mathoverflow.net/users/16447 | What is known about Ulam's problem of metric spaces with isometric squares? | In this paper:
On the uniqueness problem for metric products, Glasnik Mat. 27(47) (1992), 145-158.
By Maria Moszynska (which seems to be available for free in its entirety from google books), she notes that for compact connected metric space the result is equivalent to the analogous question for direct sums of subse... | 10 | https://mathoverflow.net/users/11142 | 77461 | 46,803 |
https://mathoverflow.net/questions/77462 | 4 | I want to find a comprehensive reference on general linear groups, which has depth discussion about its subgroups (like solvable subgroups, Abelian subgroups, and so on). Can anyone help me with this? Thanks a lot.
| https://mathoverflow.net/users/18356 | Reference for general linear groups | Suprunenko, "Matrix groups", [http://www.amazon.com/Matrix-Groups-D-Suprunenko/dp/0821813412/ref=sr\_1\_10?s=books&ie=UTF8&qid=1318007030&sr=1-10](http://rads.stackoverflow.com/amzn/click/0821813412) .
| 3 | https://mathoverflow.net/users/nan | 77469 | 46,805 |
https://mathoverflow.net/questions/77414 | 2 | Let $f :\mathbb R^n\to \mathbb [0, \infty)$ be a (continuous, $C^2$, or smooth) subharmonic function with minimum value $0$. Then we know the sublevel set $f^{-1}((-\infty, c])$ is mean convex for $c > 0$. The interior minimum set $f^{-1}(0)$ has to be minimum if it's a $C^1$ submanifold. My question is it necessary a ... | https://mathoverflow.net/users/16750 | Minimum set of subharmonic function in $\mathbb R^n$ | According to the following, no (you can make the function nonnegative by taking maximum of $f$ and the constant 0):
MR1173388 (93h:31003)
Armitage, D. H.(4-QUEEN)
Cones on which entire harmonic functions can vanish.
Proc. Roy. Irish Acad. Sect. A 92 (1992), no. 1, 107–110.
31B05
Suppose that $L$ and $M$ are two lin... | 2 | https://mathoverflow.net/users/14493 | 77473 | 46,808 |
https://mathoverflow.net/questions/77468 | 1 | Let $S\_{n,k}$ be the set of all numbers that can be written as the product of $n$ odd primes plus $2k$. Is there integers $n>1$ and $k>1$ such that $S\_{n,k}$ contains finite number of primes?
| https://mathoverflow.net/users/18359 | Primes of the form $p_{i_1}p_{i_2}\cdots p_{i_n}+2k$ | The answer to you question is very likely, no, these sets are always infinite. However, this is open. If you slightly but significantly modify your question it is however known.
More details:
I believe that it would be widely believed that all $S\_{n,k}$ contain infinitely many prime numbers. Yet, that this is unpr... | 3 | https://mathoverflow.net/users/nan | 77478 | 46,810 |
https://mathoverflow.net/questions/76166 | 0 | Hi,
Suppose you are given a dcpo of categories, where the functor is taken as the ordering relation. This collection is a category too and it has limits, or perhaps cartesian products. This comes from the shape of what the supremum looks like. This dcpo might also be a topos. Does anyone have any thoughts on this?
A... | https://mathoverflow.net/users/10007 | a dcpo of categories has limits and compact categories | To answer your more pointed form of the question: if $P$ is a poset viewed as a category then the limit of a functor $F:I \to P$ is the same thing as $\inf \lbrace F(i) : i\in I \rbrace$ and the colimit is the same thing as $\sup \lbrace F(i) : i\in I\rbrace$. If I'm not mistaken a directed complete partial order is ju... | 3 | https://mathoverflow.net/users/1148 | 77493 | 46,822 |
https://mathoverflow.net/questions/77498 | 5 | In the category Set of sets and functions, consider the functor F(X) = X \* X where \* is the product (its action on arrows is just F(f) = f \* f). Does this functor preserve pushouts? Or at least pushouts of pairs of epimorpisms?
| https://mathoverflow.net/users/7674 | The "binary" product preserves pushouts? | General pushouts, no; pushouts of pairs of epis, yes. In fact, yes to pushouts of pairs where only one of the arrows of the pair is epi.
It's very easy to see the answer is no for general pushouts: since a coproduct $A + B$ is the pushout of a pair $A \leftarrow 0 \to B$, and since the squaring functor preserves $0$... | 14 | https://mathoverflow.net/users/2926 | 77501 | 46,824 |
https://mathoverflow.net/questions/77523 | 6 | I decided to [cross-post](https://math.stackexchange.com/questions/70619/submatrix-with-only-0-or-1-entries) the question here from math.stackexchange.com because I got no answer from there.
It is a quick question on bipartite Ramsey numbers (I'm not an expert on the subject, so perhaps the question is trivial).
Wh... | https://mathoverflow.net/users/12875 | 3x3 submatrix with only $0$ or $1$ entries | According to this:
MR1622032 (99c:05139)
Hattingh, Johannes H.(SA-RAND); Henning, Michael A.(SA-NTL2)
Bipartite Ramsey theory. (English summary)
Util. Math. 53 (1998), 217–230.
the answer is $17$, not $15.$
**Addition** A proof is contained in Irving's paper "A bipartite Ramsey problem and the Zarankiewicz numbers... | 7 | https://mathoverflow.net/users/11142 | 77524 | 46,832 |
https://mathoverflow.net/questions/77465 | 1 | I have a set of N, N>>n. n-dimensional vectors and would like to represent each of them, with approximation, as a linear combination of m, m < n, n-dimensional vectors. How should I choose the vectors so that the approximation is the best possible? I think it is an extension of the least square fitting of a line throug... | https://mathoverflow.net/users/18358 | hyperplane least square through points | Hmm, I'm not sure, whether I got your problem right, but well, I'll give it a try.
I assume the N datvectors as rowvectors consisting of n datapoints (where n << N). Then the usual PCA assumes the columns as axes in an n-dimensional coordinate-system having N vectors beginning at the origin, say the data-matrix $\sm... | 1 | https://mathoverflow.net/users/7710 | 77525 | 46,833 |
https://mathoverflow.net/questions/77131 | 5 | Can you prove the following using techniques from convex analysis or linear algebra? I was originally seeking an elementary proof, but I think it is better to broaden the scope for this bounty question.
>
> A psd matrix of the form $\left(\begin{smallmatrix}
> a & b & c\\
> b & c & d \\
> c & d & e
> \end{smallmat... | https://mathoverflow.net/users/nan | Convex Analytic or linear algebraic proof that a certain psd matrix is a sum of rank 1 psd matrices | $\newcommand{\R}{\mathbb{R}}$
Let $K$ be a compact convex body in $\R^n$, or some other $n$-dimensional vector space or affine space. Then every point $p \in K$ has an extremality rank, which is the largest dimension of a flat open ball $B$ such that $p \in B \subset K$. The 0-extremal points are thus the usual extrema... | 6 | https://mathoverflow.net/users/1450 | 77544 | 46,843 |
https://mathoverflow.net/questions/72217 | 8 | tl;dr: Is there an accepted or proposed term for a topological space whose $T\_0$ quotient is sober?
The condition that a topological space be sober (and therefore equivalent to a locale) may be broken into two parts:
* there are enough points,
* and there aren't too many.
The condition that there are enough poin... | https://mathoverflow.net/users/8508 | Sober except not $T_0$? | Well, I've decided to go ahead and use ‘with enough points’. There are a lot of reasons to restrict to $T\_0$ spaces, over and above reasons to restrict to sober spaces, and at least within that context having enough points is perfectly symmetric between topological spaces and locales.
| 1 | https://mathoverflow.net/users/8508 | 77552 | 46,845 |
https://mathoverflow.net/questions/77536 | 4 | Hello, I can not figure out why a ring that is not IBN (invariant basis number) must be SBN (single basis number). More precisely: Let $R$ be a ring (with unit, generally non-commutative) such that the free $R$-module $R^n$ is isomorphic to another free $R$-module $R^m$, where $n, m$ are different natural numbers. How ... | https://mathoverflow.net/users/18376 | SBN and IBN rings | It seems that the implication does not hold. Thanks to Lukas Vokrinek for noticing this: According the example "Tom Leinster (mathoverflow.net/users/586), when is A isomorphic to A^3?" there is an abelian group $A$ such that $A$ is isomorphic to $A^3$ but not to $A^2$. Now, let $R=End(A)$. Then R is isomorphic to $R^3$... | 3 | https://mathoverflow.net/users/18376 | 77555 | 46,847 |
https://mathoverflow.net/questions/77547 | 0 | The Hermite-grand-conjecture implies that f(k)=(2^(2^5^11^(7k+1))+1)/3 is prime for all natural numbers $k$.
Is there any explicit formula that has so far been proven to produce primes for all natural numbers?
If not, is there under some reasonable restriction of closed-form formula a possible non-constructive proo... | https://mathoverflow.net/users/18381 | Formula with prime-density 1 in the integers | It depends a bit on what you accept as "explicit". E.g., there is a positive real number $A$ such that the integer part of $A^{3^n}$ is prime for all positive integers $n$. See [Wikipedia on Mills' constant.](http://en.wikipedia.org/wiki/Mills%27_constant)
| 3 | https://mathoverflow.net/users/3684 | 77559 | 46,850 |
https://mathoverflow.net/questions/41698 | 25 | Does there exist $c>0$ such that among any $n$ positive integers one may find $3$ with least common multiple at least $cn^3$?
UPDATE
Let me post here a proof that we may always find two numbers with lcm at least $cn^2$. Note that if $a < b$, $N$=lcm$(a,b)$, then $N(b-a)$ is divisible by $ab$, hence $N\geq ab/(b-a)$... | https://mathoverflow.net/users/4312 | triple with large LCM | See [my post on AoPS](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=436816)
Edit: OK, reposting here.
The first step toward the solution, as it often happens, is to generalize the problem. Instead of just one set $A$, we shall consider $3$ sets $A,B,C$ of cardinalities $|A|,|B|,|C|$ and will try to... | 12 | https://mathoverflow.net/users/1131 | 77560 | 46,851 |
https://mathoverflow.net/questions/77562 | 4 | Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x\_0,\ldots,x\_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L\_1,\ldots,L\_k\in\mathbb{P}$ of codimension $r\_i\ge 2$, respectively. Let $\pi:X\to\mathbb{P}$ be the composition of blowing up in $L\_1$, then blowing up in the strict transform of $L\_2$... | https://mathoverflow.net/users/9947 | Ample divisors on blown-up projective space | Your question is actually far too general, so let me assume $n=2$. Also in this case, there are only partial results.
In the case where all $a\_i$ are equal to $1$, Kurchle and (independently) Xu showed that
$$H=\pi^\*(dL) - \sum\_{i=1}^r E\_i$$
(where $L$ is the class of a line) is ample, provided that $H^2 > 0$ a... | 8 | https://mathoverflow.net/users/7460 | 77565 | 46,853 |
https://mathoverflow.net/questions/77569 | 2 | I have a graph $G(V,E)$ and a tree $T(V',E')$ where $|V|=|V'|$ and $T$ is isomorphic to a subgraph of $G$. In other words I found a spanning tree of $G$ and made one of its nodes act as the root.
I now have 2 problems I want to look at:
1. If I remove a node from $G$ what is the optimal way to determine if a new $T... | https://mathoverflow.net/users/14534 | Tree graph restructuring. | I think the following paper will probably answer your first question
Henzinger, M., & Valerie, K. (1997). Maintaining minimum spanning trees in dynamic graphs. *Automata, Languages and Programming, 1256*, 594-604. Doi: 10.1007/3-540-63165-8\_214
The answer to your second question can be found in several places. The... | 4 | https://mathoverflow.net/users/18372 | 77572 | 46,855 |
https://mathoverflow.net/questions/77546 | 13 | Let $X$ be a (separated, and with whatever other tameness conditions are appropriate) scheme over the integers $\mathbb{Z}$. (If you don't like schemes much, imagine that $X$ is described by algebraic equations over the integers, so that it is possible to make the algebraic variety $X(k)$ for any field $k$.) I am inter... | https://mathoverflow.net/users/1450 | Schemes over ℤ with a “graded existence over ₁” | I think if you want a good answer to your question you will have to consult papers by people who don't think F\_1 is a scam! To be more precise: since F\_1 doesn't actually exist, what do people think it is? And I think most people would say words like: there should be some category of schemes over Z which are somehow ... | 14 | https://mathoverflow.net/users/431 | 77574 | 46,856 |
https://mathoverflow.net/questions/19895 | 24 | I can construct a finitely presented group $G$ with the following property (which I use to construct something else).
>
> Given a finitely preseted group $\Gamma$, there is a subgroup $G'\le G$ of finite index
> such that $$\Gamma=G'/\langle\mathrm{Tor}\, G'\rangle ,$$ where $\mathrm{Tor}\, G'\subset G'$ is the set... | https://mathoverflow.net/users/1441 | Universal group? | Аnswered to move the question to *answered status*.
We decided to use the term **telescopic action**.
Thank you all for your comments they were helpful for me and Dima.
| 6 | https://mathoverflow.net/users/1441 | 77575 | 46,857 |
https://mathoverflow.net/questions/77573 | 2 | I trying to study the Coding Lemma (in descriptive Set Theory) and there is a small point in the proof that I don't understand. Let me first recall the version I'm studying ( there are different version of the Moschovakis Coding lemma).
Assume **AD**. Let $\Gamma$ be a non-self-dual pointclass closed under real quant... | https://mathoverflow.net/users/3859 | Moschovakis Coding Lemma | I might be missing something, but I believe the answer is the following. First suppose $\Gamma$ isn't present. Then this just amounts to showing that, if $R\subseteq X\times Y$, $Z$ is a cofinite subset of $X$, and I have a choice set $C$ for the induced $R'\subseteq Z\times Y$ (that is, $R'=R\cap (Z\times Y))$, then I... | 4 | https://mathoverflow.net/users/8133 | 77576 | 46,858 |
https://mathoverflow.net/questions/77564 | 3 | I call an H-machine a machine that can be connected to turing machines and that takes as input a natural integer n and instantly returns the n'th digit of the mathematical constant H.
Is there a proof/disproof/(or any kind of argument of what we should expect) that access to any combination of the e,Pi,sqrt(2),zeta(3... | https://mathoverflow.net/users/18381 | Hermit H-machines | Your $H$-machine concept is essentially the same as the concept of [oracle computation](http://en.wikipedia.org/wiki/Oracle_machine), due originally to Turing, which gave rise to the elaborate theory of [Turing degrees](http://en.wikipedia.org/wiki/Turing_degrees) in computability theory. The idea in that subject is th... | 7 | https://mathoverflow.net/users/1946 | 77577 | 46,859 |
https://mathoverflow.net/questions/77583 | 8 | A group is residually nilpotent if the intersection of the terms in its lower central series is the trivial group.
Is the free product of arbitarily (possibly infinite) many abelian groups residually nilpotent? In particular, this would imply that the free product of abelian groups with any free group is residually n... | https://mathoverflow.net/users/nan | Is the free product of arbitrarily many copies of `${\mathbb{Z}}$` and `${\mathbb{Z}}/2$` residually nilpotent? | The group $PSL(2,\mathbb{Z})=\mathbb{Z}/2\mathbb{Z} \* \mathbb{Z}/3\mathbb{Z}$ is not residually nilpotent. Indeed, if it was residually nilpotent, then it would be residually (finite $p$-)group since every finitely generated nilpotent group is residually finite (Malcev) and every finite nilpotent group is a direct pro... | 17 | https://mathoverflow.net/users/nan | 77592 | 46,863 |
https://mathoverflow.net/questions/76885 | 5 | Let $S$ be a hyperbolic surface, which is not the punctured torus or $4$-holed sphere. I am interested in finding a ``geometrically optimal'' pants decomposition on $S$.
Here is a candidate definition. Given a pants decomposition $P$, order the curves of $P$ *from longest to shortest* (in the hyperbolic metric). The... | https://mathoverflow.net/users/8183 | Optimal pants decompositions of a hyperbolic surface | A pair of optimal pants decompositions $A, B$ need not have disjoint shortest curves. Here is an example.
Let $S$ be the genus two hyperbolic surface built from four equilateral right-angled hexagons by "doubling". That is, let $H$ be such a hexagon. Let $a = 2 \cosh^{-1}(\sqrt{3/2}) = 1.31695\ldots$ denote the side... | 4 | https://mathoverflow.net/users/1650 | 77597 | 46,866 |
https://mathoverflow.net/questions/77594 | 12 | Let $f:X\longrightarrow Y$ be a finite morphism of schemes of degree $n$. Let $S\to Y$ be a morphism of schemes.
Is the degree of the finite morphism $X\times\_Y S \longrightarrow S$ equal to $n$?
If not, what conditions should be put on $X$ and $Y$?
If it helps, you can assume all the schemes to be integral.
| https://mathoverflow.net/users/18393 | Is the degree of a finite morphism stable by base change | I shall assume that $X,Y$ are integral, locally noetherian schemes and that $f$ is dominant. Then the degree of $f$ is the degree of the corresponding extension of fields, namely
$$deg(f)=[Rat(X):Rat(Y)]$$.
We have for the fibers $X\_y \; (y\in f(X))$ of $f$ the interesting result:
$$dim\_{\kappa (y)} \mathcal O(X\_y... | 22 | https://mathoverflow.net/users/450 | 77598 | 46,867 |
https://mathoverflow.net/questions/77545 | 3 | I want to see the following thing:
$\ \ $If $X$ is a smooth geometrically connected scheme over a field $k$ of characteristic 0, $U\subseteq X$ is a non-empty open, $(E,\nabla)$ is an integrable connection on $X/k$, then $H\_{DM}^0(X/k,(E,\nabla))\hookrightarrow H\_{DM}^0(U/k,(E,\nabla)|\_{U})$ is an isomorphism.
Her... | https://mathoverflow.net/users/18380 | Base change for the Gauss-Manin sheaf | Here is a proof of your claim:
We want to show that any flat section of $E|\_U$ extends to a flat section
of $E$. Since the extension is unique if it exists, the question is local
on $X$ so we may assume that the bundle $E = \mathcal{O}\_X^n$. The connection is
then given an $n\times n$ matrix $A$ of $1$-forms on $X$... | 3 | https://mathoverflow.net/users/519 | 77599 | 46,868 |
https://mathoverflow.net/questions/77600 | 3 | Let $X$ be a compact connected Riemann surface of genus $g>0$.
Suppose that $X$ can be defined over a number field (as an algebraic curve). Then, is it clear that each Weierstrass point of $X$ is algebraic?
| https://mathoverflow.net/users/18393 | Are Weierstrass points algebraic | Edit: as pointed out by Felipe below, what used to be here was sort of wrong-headed, but I can't delete this answer since it has already been accepted. Anyway you should do what he says: the divisor of all Weierstrass points counted with weights is defined by the vanishing of the Wronskian determinant of a basis of the... | 5 | https://mathoverflow.net/users/1310 | 77602 | 46,869 |
https://mathoverflow.net/questions/77596 | 4 | While thinking about monads in the theory of denotational semantics, I have made an observation about the Kleisli category that I would like to check
Suppose $F : \mathcal D \to \mathcal C$, $G : \mathcal C \to \mathcal D$, is an adjunction, so that $T = GF : \mathcal D \to \mathcal D$ is a monad. There is an operato... | https://mathoverflow.net/users/3676 | Inverse of Kleisli star, or "extension operator" | Sure, you could put it that way. I think it's simpler to say that there is a natural bijection between arrows $f: x \to GFy$ in $\mathcal{D}$ and arrows $g: Fx \to Fy$ in $\mathcal{C}$, because that is obvious from the hom-formulation of adjunctions:
$$\hom\_{\mathcal{C}}(Fx, Fy) \cong \hom\_{\mathcal{D}}(x, GFy).$$... | 7 | https://mathoverflow.net/users/2926 | 77603 | 46,870 |
https://mathoverflow.net/questions/77588 | 10 | Hi there,
Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "Partial Differential Equations for Scientists and Engineers". Basically in one of the chapters he describes a way one ... | https://mathoverflow.net/users/62607 | Use of games to approximate solutions to Partial Differential Equations | The connection between random walks, diffusion, and the heat equation is an amazing example of "the unreasonable effectiveness of mathematics." However, it's important to understand that this doesn't extend to other PDE's.
I'd encourage you to make the effort to dig a little deeper so that you understand why a Monte... | 12 | https://mathoverflow.net/users/9022 | 77607 | 46,871 |
https://mathoverflow.net/questions/77604 | 5 | Let $n$ be a positive integer.
Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O\_{K}^\ast$$ is at least $n$? Can we write down such a number field explicitly?
I know that the number of solutions is always finite in a fixed number field.
| https://mathoverflow.net/users/18393 | For any $n$, does there exist a number field with at least $n$ solutions to the unit equation | A slightly more general form of the above mentioned lemma states: whenever $m$ has at least two distinct prime factors and $\zeta\_m$ is a primitive $m$-th root of unity, $1-\zeta\_m$ is a unit in $\mathbf Z[\zeta\_m]$.
Choosing $a=1-\zeta\_m$ and $b=\zeta\_m$ for the various primitive roots of unity, we get $\varphi... | 14 | https://mathoverflow.net/users/2035 | 77608 | 46,872 |
https://mathoverflow.net/questions/62236 | 11 | Every curve of genus $g\leq 2$ has non trivial automorphisms. So the fiber of the forgetful morphism $\pi:\bar M\_{g,1}\rightarrow\bar M\_{g}$ over $[C]$ is not isomorphic to $C$ but to $C/Aut(C)$ (I am considering the coarse moduli space instead of the stack).
The general fiber of $\pi:\bar M\_{1,2}\rightarrow\bar M... | https://mathoverflow.net/users/14514 | Moduli of pointed Curves | The moduli space $\overline{M}\_{1,2}$ can not be the blow up of $\mathbb{P}^{2}$ in $2$ points because its rational Picard group has rank $2$, indeed it is generated by the divisor parametrizing genus $0$ irreducible nodal curves with $2$ marked points and the divisor parametrizing reducible curves whose components ar... | 6 | https://mathoverflow.net/users/14514 | 77613 | 46,874 |
https://mathoverflow.net/questions/67676 | 5 | The moduli space of stable $2$-pointed genus $1$ curves has cyclic quotient singularities, so any Weil divisor on $\overline{M}\_{1,2}$ is $\mathbb{Q}$-Cartier.
**Is one of the two boundary divisors $\Delta\_{irr}$ and $\Delta\_{0,2}$ of $\overline{M}\_{1,2}$ Cartier ?**
| https://mathoverflow.net/users/14514 | Cartier divisors on the moduli space of two pointed elliptic curves | The singularities of $\overline{M}\_{1,2}$ are located as follows:
a singularity of type $\frac{1}{4}(2,3)$ representing an elliptic curve of Weierstrass representation $C\_{4}$ with marked points $[0:1:0]$ and $[0:0:1]$;
a singularity of type $\frac{1}{3}(2,4)$ representing an elliptic curve of Weierstrass represe... | 2 | https://mathoverflow.net/users/14514 | 77615 | 46,875 |
https://mathoverflow.net/questions/77642 | 0 | In "Computing Grobner Fans" by Fukuda/Jensen/Thomas on page 2210 in Table 1 are the numbers (1,20,120,300,330,132) for some statistics on Grobner fans for Grass(2,5). This is a vector found in A126216, A033282, and A133437 of <http://oeis.org/> related to Stasheff associahedra (also to Lagrange inversion, Dyck paths, a... | https://mathoverflow.net/users/12178 | Grobner fan linked to associahedra? | This paper is about Grobner fans and uses a Stasheff associahedron in an example.
<http://www.springerlink.com/content/y102175023224321/fulltext.pdf>
| 1 | https://mathoverflow.net/users/17806 | 77648 | 46,887 |
https://mathoverflow.net/questions/77655 | 0 | Considered the following inner products:
$(1)$ $\langle x,y \rangle = \sum\_{t=1}^{n}x\_{t}y\_{t}$
$(2)$ $\langle x,y \rangle\_{c} = \sum\_{t=1}^{n}x\_{t}\bar{y}\_{t}$
consider the following surfaces:
$\underline{Surface (a)}$: $\langle x, x \rangle = 1$
$\underline{Surface (b)}$: $\langle x, x \rangle = \mat... | https://mathoverflow.net/users/10035 | Number of points on a complex sphere with pairwise inner product restriction | It's easy to see you can't have infinitely many points: there would be two that are within $\epsilon>0$ of each other, and thus would have inner product very close to $\langle z,z\rangle=1$ (which would then not be purely imaginary).
Since $\Re\langle u,v\rangle$ forms a genuine inner product on $\mathbb C^n$, two ve... | 1 | https://mathoverflow.net/users/35353 | 77656 | 46,888 |
https://mathoverflow.net/questions/77616 | 20 | I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When I was doing a perusal on "A primer of algebraic D-modules by S. C. Coutinho" the justification on the importance of D-modu... | https://mathoverflow.net/users/1997 | D-modules and Algebraic Solutions of PDEs | Perhaps you are looking for something deeper, but right there at the beginning of Hotta, Takeuchi, and Tanisaki's book on D-mods in the introduction is the connection to Linear PDEs.
I quote:
>
> Therefore, systems of linear partial differential equations can be identified with the
> D-modules having some finit... | 11 | https://mathoverflow.net/users/348 | 77659 | 46,890 |
https://mathoverflow.net/questions/77650 | 2 | Consider the following integral equation
$\phi(x) = f(x) + \frac{1}{x}\int\_0^x N(x,y)\phi(y)\;dy$,
where $f$ and $N$ are continuous and bounded functions. Are solutions $\phi$ of the above equation unique? If so, can one get an estimate of the form
$\sup\_{(0,x)} |\phi| \leq C \sup\_{(0,x)}|f|$ ?
Additional i... | https://mathoverflow.net/users/18406 | A Volterra-type equation | Not in general, since if $N(x,y)=a>1$ then the equation with $f=0$ has the solution $\phi(x)=x^{a-1}$. If, however, $N(0,0)<1$ (assuming as in the question that $N$ is continuous) then one can use the Banach fixed-point theorem (on short intervals) to get a unique solution.
| 5 | https://mathoverflow.net/users/18410 | 77661 | 46,891 |
https://mathoverflow.net/questions/77644 | 12 | Let $X$ be a scheme of finite type over $\mathbb{Z}$. Let $R$ be the ring of algebraic integers. My intuition is that $X(R)$ is practically always infinite.
More specifically, suppose that $X$ is faithfully flat over $\mathbb{Z}$, of relative dimension $\geq 1$, and the generic fiber is geometrically irreducible. Is... | https://mathoverflow.net/users/297 | Are there nonobvious cases where equations have finitely many algebraic integer solutions? | This is basically true, in view of a density theorem due to Robert Rumely
(*Arithmetic over the ring of all algebraic integers*, J. reine u. angew. Math. **368**, 1986, p. 127-133). It relies on Rumely's capacity theory, and his extension of the theorem
of Fekete-Szegö.
For a generalization, and an algebraic proof, s... | 12 | https://mathoverflow.net/users/10696 | 77670 | 46,896 |
https://mathoverflow.net/questions/77653 | 4 | Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I mean an element $e\ne 0$ such that for every positive integer $n$ there is an element f such that $e=nf$.
As Darji poi... | https://mathoverflow.net/users/5229 | A question about the additive group of a finitely generated integral domain | The answer is no in general ($e$ needs not to be in $\mathbb Z$), but one can show that $e$ is divisible in $R/\mathbb Z$ if and only if $e\in \mathbb Q\cap R$.
First let $R=\mathbb Z[1/p]$ for some prime number $p$. Then I claim that $1/p$ is divisible in $R/\mathbb Z$. Indeed for any $n\ge 1$, write $n=p^rm$ with ... | 4 | https://mathoverflow.net/users/3485 | 77680 | 46,901 |
https://mathoverflow.net/questions/77683 | 2 | Imagine I have an unknown (undirected) tree graph, $G$, with some unknown number of nodes $||V||$. However, I know the edge-length between nodes is of fixed size, $L\_{edge} = 1$, and I have access to the set of distances $(d\_1, ..., d\_i, ..., d\_M)$ between a root node, $v\_{root}$, and leaf nodes, $l\_i$.
When i... | https://mathoverflow.net/users/17193 | Recoving an unknown tree graph with knowledge of root node to leaf node distances | I think the question is a bit vague. Anyhow. You cannot recover the isomorphism type of the tree and you cannot recover the number of nodes. There is a unique tree having that set of distances with the minimal number of nodes. Namely, list the distances in increasing order
$d\_1,\dots,d\_M$. Let $r$ be an end point of ... | 4 | https://mathoverflow.net/users/7743 | 77685 | 46,902 |
https://mathoverflow.net/questions/77649 | 2 | Let us fix a representation $\pi\_\infty$ of GL(n,$\mathbb R$).
Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A\_{finite})$.
$$K=\Pi\_{v<\infty}K\_v$$
$K\_v$ is $GL(n,\mathbb Z\_v)$ for almost all v.
For automorphic forms of ($\chi$, K), we require that
for $\phi: GL(n,\m... | https://mathoverflow.net/users/2666 | Different cuspidal automorphic representations with same representations at infinity | This is precisely the content of Harish-Chandra's theorem ("Automorphic forms on Semisimple Lie Groups", LNM 68, 1968), proven for general reductive groups:
Fix a finite-dimensional representation $\delta$ of $K\_\infty$, an ideal $J$ of finite co-dimension in ${\mathcal Z}({\mathfrak g})$, a compact open subgroup $L... | 6 | https://mathoverflow.net/users/6753 | 77695 | 46,906 |
https://mathoverflow.net/questions/75381 | 10 | Ozsváth and Szabó proved Symplectic Thom conjecture [Annals of Mathematics, 151(2000), 93-124]. It states: An embedded symplectic surface in a **closed**, symplectic 4-manifold is genus-minimizing in its homology class.
Does a suitable relative version of above hold true ? More specifically, suppose we start with an ... | https://mathoverflow.net/users/5538 | Relative version of Symplectic Thom conjecture. | I think this must be a consequence of the version of the slice-Bennequin inequality proved by Mrowka and Rollin (but I might be wrong). Perhaps the argument also requires the boundary to have the "strong filling" property (Stein near the boundary).
Given a Legendrian knot $L$ in (to start with) the 3-sphere $S^3$, th... | 13 | https://mathoverflow.net/users/16193 | 77697 | 46,908 |
https://mathoverflow.net/questions/77694 | 5 | We know by the standard Implicit Function Theorem that
>
> If $f:\mathbb R^4\rightarrow\mathbb
> > R^2$ is a polynomial (or in fact any
> continuously differentiable function),
> then there is a point $a\in\mathbb
> > R^2$ such that $f^{-1}(a)$ is at least
> two-dimensional.
>
>
>
Now imagine that instead ... | https://mathoverflow.net/users/5572 | implicit function theorem for algebraic sets | This is a well-known theorem. I imagine it has been routinely used for many years, so tracking down a historical reference would be hard. The following argument will work for any $f$ which is definable over the real field, i.e. whose *graph* is a semi-algebraic set.
The graph $\Gamma \subset A\times B$ of your mappin... | 3 | https://mathoverflow.net/users/8212 | 77698 | 46,909 |
https://mathoverflow.net/questions/77635 | 70 | Maybe it would be helpful for me to summarize the little bit I
think know. A 2D CFT assigns a Hilbert space ${\cal H}$ to a circle and
an operator
$$A(X): {\cal H}^{\otimes n}\rightarrow {\cal H}^{\otimes m}$$
to a Riemann surface $X$ with $n$ incoming boundaries and $m$
outgoing boundaries. This data is subject to nat... | https://mathoverflow.net/users/1826 | What exactly is the relation between string theory and conformal field theory? | One must distinguish between quantum/classical on the string world-sheet and in spacetime.
Both of your statements are basically correct, but should read something like "CFT theory is the space of classical solutions to the spacetime equations of string theory" and "Quantization of the
the world-sheet sigma model of a ... | 27 | https://mathoverflow.net/users/10475 | 77703 | 46,911 |
https://mathoverflow.net/questions/77699 | 2 | Let $A,B$ be subfactors of a II$\_1$ factor $M$ with $A\*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a free complement for $A$ in $M$.
My first question is simple (to state): if a free complement exists is it necessarily unique? My intuition... | https://mathoverflow.net/users/10779 | Uniqueness of free complements | If you want uniqueness up to equality, then the answer is no since if $B$ is a free complement for $A$ in $M$ then so is $u B u^\*$ for any unitary $u \in A$. This is the case also in the specific example you give above.
If you want uniqueness up to isomorphism then this is more subtle since showing that II$\_1$ fact... | 5 | https://mathoverflow.net/users/6460 | 77706 | 46,913 |
https://mathoverflow.net/questions/77688 | 2 | Let $(X,\omega)$ be a compact Kahler manifold, and let $\alpha$ and $\beta$ be smooth $(1,1)$-forms on $X$ that are harmonic (with respect to $\omega$). I can consider each of my $(1,1)$-forms as an antilinear vector bundle morphism $T\_X \to \overline T\_X^\*$. Then I can fabricate a new $(1,1)$-form on $X$ by setting... | https://mathoverflow.net/users/4054 | Is a certain composition of harmonic forms again harmonic? | By your last sentence, this is clearly not true in general.
Let $X$ be a $K3$ surface and let $\omega$ be a Kähler form on $X$ whose associated metric is Ricci flat. Let $H$ denote the space of the real-valued harmonic $(1,1)$-forms $\alpha$ that satisfy $\omega\wedge\alpha = 0$, i.e., the primitive $(1,1)$-forms. T... | 8 | https://mathoverflow.net/users/13972 | 77707 | 46,914 |
https://mathoverflow.net/questions/72776 | 4 | **SECOND EDIT**: This question is now essentially answered. See [this blog entry](http://wp.me/p1mOPD-7y) for details. Thanks to everyone who commented and answered here.
---
**EDITED TO ADD**: I asked a (hopefully more pointed and understandable) [version of this question](https://cstheory.stackexchange.com/q/78... | https://mathoverflow.net/users/9197 | Is there an official name for this prohibited word pattern? | I found a solution in the literature ([Latin Squares which Contain no Repeated Digrams](http://www.jstor.org/stable/2027267) by E.N. Gilbert, 1965), obtained by producing a special kind of Latin square (an *addition square*) using two permutations that are [Costas arrays](http://en.wikipedia.org/wiki/Costas_array). Ple... | 2 | https://mathoverflow.net/users/9197 | 77708 | 46,915 |
https://mathoverflow.net/questions/77715 | 6 | Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action.
What does this imply for $M$? What are examples except for (products of) spheres?
| https://mathoverflow.net/users/3816 | $S^1$-action in three dimensions | You will find the complete answer in the papers
[Frank Raymond, Classification of the actions of the circle on 3-manifolds. Trans. Amer. Math. Soc. 131 1968 51–78.](http://www.ams.org/mathscinet-getitem?mr=219086)
and
[Peter Orlik and Frank Raymond, Actions of SO(2) on 3-manifolds. 1968 Proc. Conf. on Transforma... | 11 | https://mathoverflow.net/users/1335 | 77719 | 46,918 |
https://mathoverflow.net/questions/77713 | 2 | Hello,
I have a question about certain immersions. If $X$ is a base scheme, and $E$ is a loccally free quasi coherent sheaf over X, then we can build $P(E)$ the projective bundle associated to E.
Now if $0\to L \to E \to F\to 0$ is an exact sequence of vector bundles, with L invertible. We get a closed immersion fo... | https://mathoverflow.net/users/18422 | Closed immersion into (relative) projective bundle. | For the fact that $\mathbf P(F)\to\mathbf P(E)$ is a closed embedding, see EGA II, 4.1.2. By 4.2.9, an $X$-morphism $f\colon T\to\mathbf P(E)$ factors over $\mathbf P(F)$ iff $(pf)^\*E\to f^\*O(1)$ factors over $(pf)^\*F$. This is equivalent to $(pf)^\*L\to f^\*O(1)$ being zero, and this again is equivalent to $f$ fact... | 1 | https://mathoverflow.net/users/2035 | 77722 | 46,919 |
https://mathoverflow.net/questions/77714 | 5 | Let $M$ be a closed $n$-dimensional Riemannian manifold with positive sectional curvature.
Let $G$ be a close subgroup of isometry group ${\rm Iso}(M)$. Suppose the action of $G$ on $M$ is not transitive, hence $M/G$ has dimension at least $1$.
By a theorem of Grove and Searle the symmetry rank $${\rm symran}(M)\le ... | https://mathoverflow.net/users/16750 | Dimension of certain subgroup of isometry group of positively curved manifold | Forgetting positive curvature, if $\dim M^n/G=k$ then by looking at the transitive action of $G$ on the principal orbit one gets a trivial bound $\dim G\le \dim O(n-k)=\frac{(n-k+1)(n-k)}{2}$. This bound is realized for $k=1$ on a round $S^n$ and $G=O(n-1)$. As the sphere is positively curved this bound is sharp.
Add... | 10 | https://mathoverflow.net/users/18050 | 77729 | 46,923 |
https://mathoverflow.net/questions/77684 | 3 | this question was asked on mathunderflow but no one gave a satisfactory answer (perhaps here it will receive more attention?)
Say that one has a morphism of projective algebraic varieties $f: X \to Y$, which is birational.
There is a pushforward of cycles morphism $f\_\\*: N\_\\*(X) \to N\_\\*(Y)$.
Now, if one coul... | https://mathoverflow.net/users/16857 | if f is birational, is the pushforward map on the numerical groups surjective? | As it has been mentioned, it is not entirely clear how one should define numerical equivalence on singular varieties. What can definitely be done (and has been done) is to define
$N^1$ as the numerical group of Cartier divisors and $N\_1$ as the group of 1-cycles modulo numerical equivalence against Cartier divisors. ... | 7 | https://mathoverflow.net/users/10076 | 77740 | 46,929 |
https://mathoverflow.net/questions/77742 | 5 | This may be a very naive question. We always hear about elliptic curves over the rational numbers, or over other arithmetically significant fields or rings.
But, are there open problems or recent fertile theories related to elliptic curves over the complex numbers or is everything considered "classical" and well known?... | https://mathoverflow.net/users/4721 | Elliptic curves over the complex numbers: everything "well known"? | Your question is very vague. I don't know of any open problem about elliptic curves over the complex numbers per se, although one could come up with some unproven identity among elliptic functions or modular functions and say it's "about" elliptic curves. Then again, I am a number theorist.
As for your more specific... | 3 | https://mathoverflow.net/users/2290 | 77756 | 46,936 |
https://mathoverflow.net/questions/77750 | 20 | A popular pair of exercises in first courses on functional analysis prove the following theorem:
>
> The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.
>
>
>
My question is, is the "only if" part of this (i.e., that the unit ball of an infinite-dimensional Banach space is ... | https://mathoverflow.net/users/1044 | Is there an infinite-dimensional Banach space with a compact unit ball? | Let me try a possible answer. Take a model of $ZF$ where the axiom of choice for a denumerable family of finite sets holds but where there is an infinite Dedekind finite set $B$ (this model can be checked to exist, for instance, [here](https://web.archive.org/web/20171102221007/http://consequences.emich.edu:80/conseq.h... | 13 | https://mathoverflow.net/users/12976 | 77762 | 46,937 |
https://mathoverflow.net/questions/77730 | 20 | Suppose that there are $n$ vertices, we want to construct a regular graph with degree $p$, which, of course, is less than $n$. My question is how many possible such graphs can we get?
| https://mathoverflow.net/users/18420 | How many $p$-regular graphs with $n$ vertices are there? | McKay and Wormald conjectured that the number of simple $d$-regular graphs of order $n$ is asymptotically
$$\sqrt 2 e^{1/4} (\lambda^\lambda(1-\lambda)^{1-\lambda})^{\binom n2}\binom{n-1}{d}^n,$$
where $\lambda=d/(n-1)$ and $d=d(n)$ is any integer function of $n$ with $1\le d\le n-2$ and $dn$ even.
Bender and Canfie... | 37 | https://mathoverflow.net/users/9025 | 77764 | 46,938 |
https://mathoverflow.net/questions/77767 | 19 | What is the determinant of the Wronskian of the functions $\{\cos\ x, \sin\ x, \cos\ 2x, \sin\ 2x,\ldots, \cos\ nx, \sin\ nx\}$? This determinant seems to be an integer, and the sequence starts with 1, 18, 86400, 548674560000... It is not in the Encyclopedia.
**Question** What is this sequence? I guess it is enoug... | https://mathoverflow.net/users/nan | The Wronskian of sin(kx) and cos(kx), k=1...n | If you only need that it doesn't depend on $x$ consider the following argument. For any $\pi \in S\_{2n}$ written as $\pi(1)\pi(2)\cdots \pi(n)$, let $[\pi]$ be the set of all permutations $\sigma \in S\_{2n}$ which satisfy $\lbrace\sigma(2k-1),\sigma(2k)\rbrace =\lbrace\pi(2k-1),\pi(2k)\rbrace$ as sets for all $k$. Th... | 10 | https://mathoverflow.net/users/2384 | 77769 | 46,941 |
https://mathoverflow.net/questions/77681 | 17 | The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter of $B$.
There are several ways to define the perimeter. Here is a unusual one: if $A \subset \mathbb{R}^2$ is a **conve... | https://mathoverflow.net/users/908 | Isoperimetric-like inequality for non-connected sets | As you noticed, it is sufficient to consider the case
$$F=\bigcup\_{i=1}^n F\_i$$
where $F\_1$, $F\_2,\dots, F\_n$ are disjoint convex figures with nonempty interior.
Let $s$ be mean shadow of $F$.
Denote by $K$ the convex hull of all $F$.
Note that
$$\mathop{\rm length}(\partial K\cap F)\le s.$$
We will prove ... | 5 | https://mathoverflow.net/users/1441 | 77784 | 46,951 |
https://mathoverflow.net/questions/77787 | 0 | I have a couple of questions regarding the list of discriminants of real quadratic fields with narrow class number 1.
The sequence A003655 in OEIS portraits a list of discriminants of real quadratic fields with narrow class number 1. In the sequence there is no indication that the list is complete. Q1: Is that the ca... | https://mathoverflow.net/users/11134 | About list of discriminants of real quadratic fields with narrow class number 1? | One of the references at your oeis sequence is
D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241. On page 103, he proves that the narrow class group you ask about is isomorphic to the class group of binary quadratic forms. On page 82, Buell points out that computations show about 80 percent of... | 4 | https://mathoverflow.net/users/3324 | 77788 | 46,952 |
https://mathoverflow.net/questions/77520 | 8 | Inspired by a [question about bisection](https://mathoverflow.net/questions/77442/) I wondered about the following: The are two players X and Y and a moderator Z who knows two (random,independent, uniformly chosen) hidden reals $x$ and $y$ from (0,1). The game has $t$ turns. At the beginning of each turn player X knows... | https://mathoverflow.net/users/8008 | A competitive root finding game | Possibly nothing simple, even when there is only one turn left. Suppose that there is just one turn left, the intervals are of length 4 and 6 (just to scale up) and the moves must create intervals of length $\frac{j}{4}.$ Then by my calculations, the player with the shorter interval should guess $$\frac54, \frac64 \tex... | 0 | https://mathoverflow.net/users/8008 | 77789 | 46,953 |
https://mathoverflow.net/questions/77796 | 6 | It seems to be known that Hirsch length and cohomological dimension agree for (torsion-free, finitely generated) polycyclic groups.
If we drop the assumption "torsion-free", then cd is of course infinite. But, is it still true (as one might expect) that the rational cohomological dimension is bounded above by the Hi... | https://mathoverflow.net/users/17204 | Hirsch length and cohomological dimension | Hillman extended the notion of Hirsch length to elementary amenable groups and proved that it is bounded above by the rational cohomological dimension. The reference is
>
> Jonathan A. Hillman, Elementary amenable groups and 4-manifolds with Euler characteristic 0, J. Austral. Math. Soc. (Series A) 50 (1991), 160-1... | 8 | https://mathoverflow.net/users/2384 | 77799 | 46,958 |
https://mathoverflow.net/questions/77795 | 3 | Is it possible to construct the Kuga-Satake abelian variety attached to a K3 surfaces (over a local field) only using p-adic methods?
If the K3 surface is defined over a local field, the Kuga-Satake abelian variety is defined over the same local field? over a finite extension?
Where can I look for work/results in... | https://mathoverflow.net/users/17495 | Kuga-Satake with p-adic methods | A good place to look is the paper "Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic". J. Reine Angew. Math. 648 (2010), 13–67, by Jordan Rizov. There are also related results by Yves Andre which are mentioned in Rizov's paper.
| 4 | https://mathoverflow.net/users/519 | 77802 | 46,959 |
https://mathoverflow.net/questions/77806 | 2 | Consider the Schrödinger operator $H\_\hbar = -\hbar^2\Delta + V$ on $M=\mathbb{R}^n$, where $V$ is a potential that behaves well in a certain sense ($C^\infty$, bounded from below, going to infinity for large $|x|$, ...).
In the case $V = x^2$, $n=1$, the eigenfunctions are $\psi\_n(x/\sqrt{\hbar})$ where the $\psi\... | https://mathoverflow.net/users/16702 | Eigenvalues in the semiclassical limit | The canonical reference is:
Introduction to spectral theory: with applications to Schrödinger operators by Hislop and Sigal.
Your statement about the semiclassical behavior of eigenvalues seems to be proved by Barry Simon in:
<http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1983__38_3/AIHPA_1983__38_3_295_0/AIHPA_1... | 3 | https://mathoverflow.net/users/11142 | 77807 | 46,962 |
https://mathoverflow.net/questions/77790 | 5 | It is well known that if a field K is quasi-algebraically closed (i.e. all forms with coefficients in K of degree d in n > d variables have a non-trivial zero in K) then it has no central divison algebras. It is also known that there are no central division algebras over the rational cyclotomic field (obtained by adjoi... | https://mathoverflow.net/users/18443 | Existence of a non-trivial zero (in the rational cyclotomic field) of a form | This is an old conjecture of Artin and, as far as I know, it is still open. It is mentioned as such on page 477 of this 2007 article on the work of Lang:
<http://www.ams.org/notices/200704/fea-lang-web.pdf>
| 4 | https://mathoverflow.net/users/2290 | 77813 | 46,966 |
https://mathoverflow.net/questions/77812 | 7 | If $F$ is any field, $\bar{F}$ its algebraic closure, then it is well-known that all irreducible (indecomposable) $\bar{F}$-representations of a finite group $G$ are realizable over some *finite* extension $E$ of $F$, and we call $E$, a splitting field for $G$. Any extension of $E$ is also a splitting field of $G$. By ... | https://mathoverflow.net/users/17456 | Uniqueness of splitting field for linear representations of finite groups | You need two conditions for a field to be a splitting field for a specific irreducible representation (in characteristic zero to begin with): It must contain the character values of the representation. For this there is of course a minimal field, the field generated by those values. However, a splitting field must also... | 17 | https://mathoverflow.net/users/4008 | 77824 | 46,972 |
https://mathoverflow.net/questions/77754 | 5 | Does anyone know of generalizations of pcf theory where we might consider products of the form:
$$\aleph\_1 \times (\aleph\_2 \times \aleph\_2) \times (\aleph\_3 \times \aleph\_3 \times \aleph\_3) \dots$$
$$(\aleph\_1 \times \aleph\_2 \times \dots) \times (\aleph\_1 \times \aleph\_2 \times \dots) \times \dots$$
o... | https://mathoverflow.net/users/7521 | Generalizations of pcf theory | I don't know if anyone has looked at such things systematically, but I know Shelah has made use of structures of this form at various times. The examples which follow are just what I can remember off-hand; I know there's more buried in his work, but this is where I remembered seeing such a construction:
1. Clause $(\... | 6 | https://mathoverflow.net/users/18128 | 77825 | 46,973 |
https://mathoverflow.net/questions/77817 | 8 | It is known that the mapping class group of a closed (orientable) surface is generated by elements of finite order. Is this also known to be true for $Out(F\_n)?$ A related question is the following: The mapping class group is known to be a perfect group for $g\geq 3,$ that is, it is equal to its commutator. Is this kn... | https://mathoverflow.net/users/11142 | generators of Out(F_n) and homology | $Aut(F\_2)$ is generated by torsion, and [$Aut(F\_n)$ is normally generated by $Aut(F\_2)$](http://en.wikipedia.org/wiki/Nielsen_transformation), so $Aut(F\_n)$ is generated by torsion, hence $Out(F\_n)$.
| 13 | https://mathoverflow.net/users/1345 | 77826 | 46,974 |
https://mathoverflow.net/questions/77823 | 0 | Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$).
Consider the following high-degree extension of $OR(x)$: Choose a prime $q\in[n^2,2n^2]$ and let $p:\mathbb{F}\_q^n\to\mathbb{F... | https://mathoverflow.net/users/nan | Low degree approximation of the polynomial extension of the logical-or function | No. In fact, there is no such polynomial of degree smaller than $n$. Otherwise, $f=p-p'$ is a nonzero polynomial of degree at most $n$, hence
$$\Pr\_{x\in\mathbb F\_q^n}(p(x)=p'(x))=\Pr\_{x\in\mathbb F\_q^n}(f(x)=0)\le\frac nq<1-\epsilon$$
by the Schwartz–Zippel lemma.
| 1 | https://mathoverflow.net/users/12705 | 77830 | 46,975 |
https://mathoverflow.net/questions/77816 | 14 | First, a disclaimer: This is a repost of [a question I asked on stackexchange](https://math.stackexchange.com/questions/71235/do-these-matrix-rings-have-non-zero-elements-that-are-neither-units-nor-zero-divi) (no answer there).
Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ ... | https://mathoverflow.net/users/17263 | Do these matrix rings have non-zero elements that are neither units nor zero divisors? | The answer given by David Speyer can be strengthened as follows. If $A$ is a non-invertible $n\times n$ matrix with entries in $R$ as described in the problem, then the linear maps $R^n \to R^n$ defined by either left or right multiplication are non-injective. In particular, $A$ is both a left-zero-divisor and a right-... | 10 | https://mathoverflow.net/users/778 | 77834 | 46,977 |
https://mathoverflow.net/questions/77786 | 11 | A recent question [Random Reidemeister moves to unknot](https://mathoverflow.net/questions/77570/random-reidemeister-moves-to-unknot) contains a link to the paper <http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf>, in which J. Hass and J. Lagarias show that one can transform a... | https://mathoverflow.net/users/2349 | Number of the Reidemeister moves needed to transform one diagram into another one | Coward and Lackenby have an [upper bound on the number of Reidemeister moves](https://arxiv.org/abs/1104.1882), which is a tower of exponentials. The existence of some such bound is not surprising, since Waldhausen had proven that the knot isotopy problem was solvable, so some computable upper bound exists.
Suppose y... | 5 | https://mathoverflow.net/users/1345 | 77837 | 46,978 |
https://mathoverflow.net/questions/77836 | 16 | This is a question about [Rubik's Cube](https://en.wikipedia.org/wiki/Rubik%27s_Cube) and generalizations of this puzzle, such as [Rubik's Revenge](https://en.wikipedia.org/wiki/Rubik%27s_Revenge), [Professor's cube](https://en.wikipedia.org/wiki/Professor%27s_Cube) or in general the $n \times n \times n$ cube.
Let $... | https://mathoverflow.net/users/2841 | God's number for the $n \times n \times n$-cube | I am not an expert, but I remember seeing [this press release from MIT](https://news.mit.edu/2011/rubiks-cube-0629) not too long ago. [The corresponding arXiv article](https://arxiv.org/abs/1106.5736) should answer your second question.
| 14 | https://mathoverflow.net/users/78 | 77841 | 46,981 |
https://mathoverflow.net/questions/77829 | 3 | Dear mathoverflow community,
I'm looking for a translation (English, French or German) of Drinfeld's paper "Coverings of p-adic symmetric domains". If there is no translation out there, maybe someone knows other sources where the content of Drinfeld's paper is covered (in one of the three mentioned languages).
Than... | https://mathoverflow.net/users/5831 | Drinfeld's "Coverings of p-adic symmetric domains" translated? | The English translation of Drinfeld's paper is available here:
<http://www.springerlink.com/content/j587364352k53717/>
| 3 | https://mathoverflow.net/users/12205 | 77843 | 46,983 |
https://mathoverflow.net/questions/76975 | 8 | Suppose we have a graph where every edge is colored red or blue. We say that a path is *alternating* if the red and blue edges alternate in it. Our goal is to find many edge/vertex-disjoint alternating paths from a given vertex $s$ to another given vertex $t$. Has this problem been studied before?
Update: It DOES NOT... | https://mathoverflow.net/users/955 | Red-blue alternating Menger's theorem | It seems that the problem of determining whether there exist $2$ vertex disjoint red-blue alternating paths joining vertices $s$ and $t$ is NP-complete. Thus, unless NP $=$ co-NP, there exist no efficient characterization of obstructions to existence of such paths, similar to the one you propose in the lemma.
Below i... | 5 | https://mathoverflow.net/users/8733 | 77848 | 46,985 |
https://mathoverflow.net/questions/77856 | 3 | *(Cross posted from [this math.SE](https://math.stackexchange.com/questions/66375/) question)*
Let $X$ be a Cech-complete space, and $Y$ a paracompact space. Suppose $f\colon X\to Y$ is a continuous and open surjection.
Since $Y$ is completely regular we have that $\beta(Y)$ is homeomorphic to $Y$ as a dense subse... | https://mathoverflow.net/users/7206 | Extending open maps to Stone-Cech compactifications | Let $Y=(-1/n)\_{n=1}^\infty \cup \{0\}$, $B$ the positive integers, $X=Y\cup B$ with the topology they inherit from the real line. Define $f:X\to Y$ to be the identity on $Y$ and $f(n)=-1/n$ for $n$ in $B$. The closure of $2B$ in $\beta X$ is open and onto $\{0\} \cup (1/2n)\_{n=1}^\infty$ in $Y$, which is not open.
... | 3 | https://mathoverflow.net/users/2554 | 77877 | 46,995 |
https://mathoverflow.net/questions/77682 | 2 | Consider the Euclidean group $E(n)$ as the semidirect product for Euclidean vector space $\mathbb{E}^n$ with its orthogonal group $O(\mathbb{E}^n)$:
$E(n)=\mathbb{E}^n\rtimes O(\mathbb{E}^n)$
Then the following short exact sequence splits
$1\rightarrow \mathbb{E}^n\rightarrow E(n)\rightarrow O(\mathbb{E}^n)\right... | https://mathoverflow.net/users/17551 | Subgroups of the Euclidean group as semidirect products | To shed light on some questions:
1. The statement "$G=T\rtimes Q \Leftrightarrow Q \cong Aut(\mathcal{L}^n)$" is wrong.
2. The statement "if $Q \cong Aut(\mathcal{L}^n)$ and $T \cong \mathcal{L}^n$, then $G=T\rtimes Q$" is wrong.
3. The following is true:
>
> Theorem: There is a 1-1 correspondence between symmor... | 5 | https://mathoverflow.net/users/10194 | 77882 | 46,998 |
https://mathoverflow.net/questions/77884 | 8 | ... every Riemann surface of genus $1$ appears as a complex one-parameter subgroup of $G$?
| https://mathoverflow.net/users/6862 | Does there exist a complex Lie group G such that ... | No. In a connected complex Lie group all compact complex subgroups must be in the center, that is, in the kernel of the adjoint representation, because in a complex general linear group there is no nontrivial compact connected complex subgroup. And in a connected abelian complex Lie group there can be only countably ma... | 16 | https://mathoverflow.net/users/6666 | 77890 | 47,001 |
https://mathoverflow.net/questions/77885 | 6 | It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations".
I wonder if there is a 2-version of this. In particular,
>
> Does a category have all finite 2-limits if it has all 2-equ... | https://mathoverflow.net/users/348 | 2-completeness analog of completeness theorem | Your suspicion is correct: in general, a V-category has all weighted V-limits if it has all conical V-limits *and* is cotensored over V (see Kelly's *Basic Concepts of Enriched Category Theory*, section 3.10). For V = Cat (and this is true for bicategories too), cotensors can be constructed from conical limits and cote... | 7 | https://mathoverflow.net/users/4262 | 77893 | 47,002 |
https://mathoverflow.net/questions/77905 | 0 | Hello, there is a statement as following:
If every point of X is a G\_delta and X is T\_1, then take Y = set of X,
plus the topology generated by all open sets needed to prove G\_delta-ness of every singleton,
plus the cofinite topology, then Y is a condensation of X (using identity) and is first countable
by cons... | https://mathoverflow.net/users/18465 | Topology generated by the collection of open sets | So we start with $(X, \mathcal{T})$, a $T\_1$ space in which every point is a $G\_{\delta}$, as witnessed by open sets $U\_n(x)$, $n \in \mathbb{N}$, $x \in X$. W.l.o.g. we can take these sets to be decreasing.
The topology generated by all these sets we call $\mathcal{T}'$, say, and it is $T\_1$, because for every $... | 2 | https://mathoverflow.net/users/2060 | 77909 | 47,006 |
https://mathoverflow.net/questions/77891 | 13 | (Disclaimer: This question was also asked at MSE (<https://math.stackexchange.com/questions/71020/can-we-collapse-omega-1-without-adding-a-dominating-real>). I'm posting it here because, when I asked it, I was torn between my sense that it was appropriate for MO and my suspicion that this question is much, much easier ... | https://mathoverflow.net/users/8133 | Can we collapse $\omega_1$ to $\omega$ without adding a dominating real? | As Amit points out in his comment, if the Continuum Hypothesis holds in the ground model $V$, then the collection of ground model reals becomes countable in the extension $V[G]$ in which $\omega\_1$ is collapsed, and therefore there must be a real in $V[G]$ dominating every real in $V$. More generally, consider the dom... | 12 | https://mathoverflow.net/users/1946 | 77922 | 47,014 |
https://mathoverflow.net/questions/77819 | 5 | $\DeclareMathOperator\Spec{Spec}$Let $L \rightarrow X$ be an ample line bundle over $X$ which is a compact complex manifold. Suppose that I have a ***first-order*** deformation of the pair $(X,L)$. When does this first-order deformation gives rise to a **complex** deformation of the pair $(X,L)$ in the sense of Kodaira... | https://mathoverflow.net/users/18450 | From first-order deformation to complex deformation of a pair $(X,L)$ | What you are typically looking for is a "true" deformation over an algebraic or analytic pointed curve $(T, 0)$ such that the tangent vector to $T$ at $0$ corresponds to the infinitesimal deformation you have.
The functor of infinitesimal deformations of a pair $(X, L)$ admits a semiuniversal formal deformation (see ... | 4 | https://mathoverflow.net/users/11528 | 77929 | 47,016 |
https://mathoverflow.net/questions/77937 | 3 | I have a signed measure $\mu$ on a convex subset $C\subset \mathbb{R}^n$, and I want to prove that $\mu$ is a probability measure, most importantly that it is positive everywhere.
I do know that $\int f(x)d\mu(x)\geq 0$ for any positive CONVEX function $f$. So if I could get this inequality for indicator functions I'... | https://mathoverflow.net/users/18474 | Signed measure that is positive over convex sets | A counterexample is a signed measure on the interval $I:=[-1,1]$ concentrated in the points $\{-1\}$, $\{0\}$, $\{1\}$ with
weights respectively $1/2$, $-1$, $1/2$. (Thus $\int\_If d\mu= f(1)/2+f(-1)/2 - f(0)\ge0$ is just the convexity inequality).
| 11 | https://mathoverflow.net/users/6101 | 77939 | 47,020 |
https://mathoverflow.net/questions/77934 | 34 | There is a trend, for some people, to study representations of quivers. The setting of the problem is undoubtedly natural, but representations of quivers are present in the literature for already >40 years.
Are there any connections of this trend with other Maths? For, it seems like it is a self-contained topic and b... | https://mathoverflow.net/users/13070 | Are quivers useful outside of Representation Theory? | In addition to being a nice example for abelian, $A\_{\infty}$ and Calabi–Yau categories, and being a prototypical example for [Generalized Donaldson–Thomas Invariants and the Wall Crossing Phenomenon](https://arxiv.org/abs/0811.2435), quivers have a lot of applications in various different fields. Since the question i... | 32 | https://mathoverflow.net/users/9534 | 77943 | 47,023 |
https://mathoverflow.net/questions/77951 | 2 | Yesterday I was wandering for the $n$-lab and I've found the definition of $n$-poset.
Following this [post](http://ncatlab.org/nlab/show/n-poset) it seems that a $n$-poset should be a $(n,n+1)$-category.
Now an $(n,r)$-category should be a category such that every $k$-morphism is an equivalence for $k\geq r$ and every... | https://mathoverflow.net/users/14969 | What are $n$-poset? | It seems to me that you are effectively asking why a $(0,1)$-category is a poset. Because if that is so, it makes sense to define an $n$-poset to be an $(n-1,n)$-category.
To see why a $(0,1)$-category is a poset, just unwind the definition: it contains possibly non-invertible 1-morphisms, but any two of them that ha... | 3 | https://mathoverflow.net/users/381 | 77955 | 47,028 |
https://mathoverflow.net/questions/48591 | 35 | A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs.
If p=1 mod 4 is a prime, we can define the Paley graph, call it $P(p)$, of order p as follows. This graph has vertex set {0, 1, 2, ...., p-1} with two vertices i... | https://mathoverflow.net/users/1579 | Cliques, Paley graphs and quadratic residues | I'm certain that no "material" improvement to the upper bound has been obtained, and this seems to be a very hard open question. Some time ago Tom Sanders showed me an argument which improves the bound in the case $p = n^2 +1$ to $n - 1$. So far as I can tell he hasn't published this. Even if you regard the $-1$ as a "... | 17 | https://mathoverflow.net/users/5575 | 77963 | 47,033 |
https://mathoverflow.net/questions/77865 | 14 | We say that a surface $f(x,y,z)=0$ is ruled if for each point $p$ in the surface there is a line that passes through $p$ and is contained in the surface. See <http://en.wikipedia.org/wiki/Ruled_surface> for more information.
Does anybody know if there is a partial differential equation whose solutions are all ruled ... | https://mathoverflow.net/users/5572 | partial differential equation for ruled surfaces | Here is a test for when a surface of the form $z = f(x,y)$, where $f$ is a sufficiently smooth function of two variables, is ruled.
To begin, set $I\!I = f\_{xx} dx^2 + 2f\_{xy}dxdy + f\_{yy}dy^2$. If $I\!I$ vanishes identically, then the surface is a plane, so it is ruled.
Suppose that $I\!I$ is nonzero. The *disc... | 18 | https://mathoverflow.net/users/13972 | 77967 | 47,035 |
https://mathoverflow.net/questions/77959 | 9 | Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction over $K$?
I can write down curves over *certain* number fields without semi-stable reduction, but I can't do it for a ge... | https://mathoverflow.net/users/18464 | Can we always find a curve which doesn't have semi-stable reduction | Yes. Take $f\in O\_K$ a uniformizing element of some prime $\mathfrak p$. Consider the hyperelliptic curve defined by the equation
$$y^2=x^{2g+1}+f.$$
Then this curve doesn't have semi-stable reduction at $\mathfrak p$. In fact, this equation defines a proper regular model of the curve over the localization $O\_{K, \... | 15 | https://mathoverflow.net/users/3485 | 77969 | 47,036 |
https://mathoverflow.net/questions/77932 | 1 | Let $X$ be a manifold, $i: Z\to X$ is a closed embedding.
For a sheaf $S$ (of abelian groups) on a manifold $X$ and each $\varepsilon>0$ we denote by $Z\_\varepsilon$ the set of points of $X$ that lie at distance $<\varepsilon$ from $X$. Consider the sheaf $S\_\varepsilon:U\mapsto S(U\cap Z\_\varepsilon)$, and also t... | https://mathoverflow.net/users/2191 | Restriction of a sheaf to an infinitely small neighbourhood of a closed submanifold: how to work with this ind-sheaf? | Let me show that $i^{-1}$ can't have a left adjoint when $X$ is a connected topological space and $Z\neq X$ is a point. From the remark by Denis-Charles Cisinski it would follow that $i\_\* i^{-1}$ can't have a left adjoint either.
Suppose $Z=\{x\}$ and $i^{-1}$ had a left adjoint $J$. Then we would have $$Hom (JF,G)... | 3 | https://mathoverflow.net/users/2349 | 77972 | 47,038 |
https://mathoverflow.net/questions/77986 | 19 | Pell equations can be solved using continued fractions. I have heard that some elliptic curves can be "solved" using continued fractions. Is this true?
Which Diophantine equations other than Pell equations can be solved for rational or integer points using continued fractions? If there are others, what are some good ... | https://mathoverflow.net/users/17053 | Which Diophantine equations can be solved using continued fractions? | [edited to insert paragraph on Cornacchia and point-counting]
Continued fractions, or (more-or-less) equivalently the Euclidean algorithm, can be used to find small integer solutions of linear Diophantine equations $ax+by=c$, and integer solutions of quadratic equations such as $x^2-Dy^2=\pm1$ ("Pell"). Continued fra... | 25 | https://mathoverflow.net/users/14830 | 77987 | 47,045 |
https://mathoverflow.net/questions/77992 | -1 | Is $\mathbb{C}[x,y]$ isomorphic to $\mathbb{C}[x]\otimes\_{\mathbb{R}}\mathbb{C}[y]$ as rings?
Generally, in a category $\mathcal{C}$ with fibered product, morphisms $f: X\rightarrow Z$, $g:Y\rightarrow Z$, and $h: Z\rightarrow W$. when can say $X\times\_Z Y\simeq X\times\_W Y$?
| https://mathoverflow.net/users/14854 | Is $\mathbb{C}[x,y]$ isomorphic to $\mathbb{C}[x]\otimes_{\mathbb{R}}\mathbb{C}[y]$ as rings? | Well ${\bf C}[x,y]$ is isomorphic with $({\bf C}[x]) \otimes\_{\bf C} ({\bf C}[y])$. But the tensor products **over ${\bf R}$** cannot coincide, because already ${\bf C} \otimes\_{\bf R} {\bf C}$ has zero divisors. For example, $1 \otimes 1 \neq \pm \phantom. i \otimes i$ but $(1 \otimes 1)^2 = (i \otimes i)^2$, so $1 ... | 7 | https://mathoverflow.net/users/14830 | 77995 | 47,051 |
https://mathoverflow.net/questions/77968 | 0 | Irreducible polynomials are often introduced as the analog to prime numbers in polynomial rings. Prime numbers, of course, have a very rich theory, leading to the likes of the Riemann Zeta function and the Prime Number Theorem.
Do any analogs and/or generalizations of primes, such as irreducible polynomials and prime... | https://mathoverflow.net/users/17301 | Primes are to Irreducible Polynomials as Prime-related theorems are to ?? | Maybe the first generalization of prime numbers is to prime ideals in algebraic number fields. You do get analogs of the zeta-function, the Prime Number Theorem, even the Riemann Hypothesis. Any text on algebraic number theory will take you there.
| 2 | https://mathoverflow.net/users/3684 | 77999 | 47,055 |
https://mathoverflow.net/questions/77911 | 4 | I am a graduate student planning to apply in Australia for a PHd very soon.
I have observed that most of the universities require the applicant to provide a 100-words summary of the research they propose to undertake.
Now my graduate training was fairly general (MS doesn't really help you specialize). I am very clear o... | https://mathoverflow.net/users/5679 | Level of detail on a Phd application | I'm on the Mathematics faculty of an Australian university. The way it seems to work at my university, and maybe at other Australian universities as well, is that a PhD applicant is accepted only if there is a faculty member committed to acting as supervisor. So it seems to me that the indicated strategy is, when apply... | 6 | https://mathoverflow.net/users/3684 | 78000 | 47,056 |
https://mathoverflow.net/questions/77948 | 5 | This is the little brother of [question 68071](https://mathoverflow.net/questions/68071): elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings experience will see.
Let $n$ and $m$ be positive integers. Let $k$ be a field o... | https://mathoverflow.net/users/2530 | An isomorphism of 2-Schur modules | As long as 2 is invertible, we can use the isomorphism $W^{\otimes 2} \cong S^2 W \oplus \wedge^2 W$ to split $(U \otimes V)^{\otimes 2}$ in two different ways:
1. $(S^2U \otimes S^2V) \oplus (\wedge^2 U \otimes \wedge^2 V)$ is the symmetric summand, hence isomorphic to $S^2(U \otimes V)$.
2. $(S^2U \otimes \wedge^2 ... | 5 | https://mathoverflow.net/users/121 | 78010 | 47,061 |
https://mathoverflow.net/questions/78002 | 3 | Let $X$ be an infinite dimensional vector space over a field $\mathbb{K}$. Suppose that $(X,\|\cdot\|)$ is a complete normed vector space, in the sense that any Cauchy sequence is convergent. Suppose that the closed unit ball of $X$ is compact in the strong topology.
**Question 1.**
Is $X$ necessarily isomorphic... | https://mathoverflow.net/users/2386 | Infinite dimensional vector spaces with compact unit ball | I think the following meets your setup. Let $\mathbb K = \mathbb Z\_2$ with the "absolute value" $|0|=0, |1|=1$ (this is non-Archimedean). See <http://en.wikipedia.org/wiki/Absolute_value_%28algebra%29>
Set $V = \mathbb Z\_2^I$ for some index set $I$, with the trivial norm $\|0\|=0$ and $\|x\|=1$ for all other vector... | 5 | https://mathoverflow.net/users/406 | 78020 | 47,064 |
https://mathoverflow.net/questions/78031 | 6 | Given an abelian category $\mathcal{A}$ the category of chain complexes over $\mathcal{A}$ is again an abelian category. If $\mathcal{A}$ is a Grothendieck category then the category of chain complexes over $\mathcal{A}$ is a Grothendieck category? In praticular, for a ring $R$ with unitary and the category of its left... | https://mathoverflow.net/users/8648 | About the category of chain complexes and Grothendieck categories. | Yes, this is stated e.g. on page 3 of [Hovey: Model category structures on chain complexes of sheaves](http://www.math.uiuc.edu/K-theory/0366/sheaves.pdf).
| 7 | https://mathoverflow.net/users/15887 | 78032 | 47,070 |
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