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https://mathoverflow.net/questions/49266 | 10 | This question was inspired by [this one.](https://mathoverflow.net/questions/49259/when-is-product-of-two-ideals-equal-their-intersection) Given two ideals $A,B$ in a finitely generated commutative ring $R$. Is it possible to decide whether $A\cap B=AB$? Here $R$ is given by generators and relations, i.e. as a factor-r... | https://mathoverflow.net/users/nan | intersection of ideals in a commutative ring vs their product | I think this problem can in fact be handled by Gröbner basis theory in the case $A$ is a polynomial ring. Since $I\cdot J \subseteq I\cap J$ for any two ideals, one can simply compute a Gröbner basis of $I\cap J$ (which is computed as the elimination ideal $( t\cdot I+(1-t)\cdot I ) \cap A$) and then checking whether e... | 9 | https://mathoverflow.net/users/3996 | 49271 | 31,025 |
https://mathoverflow.net/questions/49190 | 4 | Hartshorne in his chapter on surfaces defines a ruled surface(over an algebraically closed field) to be a smooth projective surface $X$ together with a surjective morphism $\pi:X\to C$, $C$ a smooth curve, such that the fiber over each point $y\in C$, call it $X\_y$, is isomorphic to $\mathbb{P}^1$, and such that $\pi$... | https://mathoverflow.net/users/11395 | Section of a Ruled surfaces | I thought I'd expand my earlier comment, which was not all that clear,
and I'm not even sure where you'd look it up.
Let's say that a ruled surface over a smooth curve $C$ is smooth projective morphism
$f:X\to C$ all of whose fibres are isomorphic to $\mathbb{P}^1$. Then one checks
that $\omega\_{X/C}^{-1}$ is relative... | 6 | https://mathoverflow.net/users/4144 | 49297 | 31,039 |
https://mathoverflow.net/questions/49259 | 60 | Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds to the union $V(I\cap J)=V(I)\cup V(J)$. But a product $I.J$ gives a new subscheme $V(I.J)$ which has same support as th... | https://mathoverflow.net/users/10408 | When is the product of two ideals equal to their intersection? | To add to David Speyer's answer, since this story continues with a rather interesting and illustrious history:
When $A$ is regular, the Tor functor satisfies the following property:
>
> (1) $\text{Tor}\_1^A(M,N) = 0$ implies $\text{Tor}\_i^A(M,N) = 0$ for $i>0$ for any two finitely generated modules.
>
>
>
(... | 36 | https://mathoverflow.net/users/2083 | 49299 | 31,040 |
https://mathoverflow.net/questions/49304 | 2 | Let $I$ be an ideal and let $I^+$ denote its complement (the so-called $I$-positive sets). Now we say that $I$ is $\lambda$-saturated iff each antichain in $I^+$ has size less than $\lambda$. Further $sat(I)$ is the least cardinal $\kappa$ such that $I$ is $\kappa$-saturated.
It can be shown that if $sat(I)$ is infin... | https://mathoverflow.net/users/4753 | Is $sat(I)$ always a regular cardinal? | You do not need the $\kappa$-completeness of $I$. In fact the following holds for an arbitrary partial order and is exercise F4 of chapter VII in Kunen's book Set Theory: an Introduction to Independence Proofs.
Theorem. (Tarski) Let $\mathbb{P}$ be a poset, and let $\kappa$ be the least cardinal for which $\mathbb{P}... | 5 | https://mathoverflow.net/users/2436 | 49309 | 31,045 |
https://mathoverflow.net/questions/49236 | 10 | Any continuous function can be uniformly approximated by smooth functions.
I would like to have something similar - in what-ever sense - for continuous manifolds.
For example, by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space ... | https://mathoverflow.net/users/2082 | A senseful meaning of 'approximation of manifolds'? |
>
> A senseful meaning of ‘approximation of manifolds’?
>
>
>
A simply-looking meaning is the one of Gromov-Hausdorff, as mentioned by Ryan Budney.
It has many cool applications, but none that I'm aware of are to topological manifolds.
The kind of approximation that one normally uses to prove something about t... | 10 | https://mathoverflow.net/users/10819 | 49316 | 31,049 |
https://mathoverflow.net/questions/49286 | 6 | Often you need a notation for a finite sequence with one element is removed;
i.e. $$(x\_1,\dots,x\_{i-1},x\_{i+1}\dots, x\_n).$$
I know one notation
$$(x\_1,\dots,\hat x\_i,\dots, x\_n)$$
and I hate it. It is too long and it has no sense; i.e., unless you know the meaning you will never guess what is it.
**Question:*... | https://mathoverflow.net/users/10330 | notation for finite sequence with one element is removed | In game theory, such sequences are needed all the time, and the notation $x\_{-i}$ has become so common that it is often not even defined in papers.
The reason is that much of game theory is concerned with situations where each player $j$ has a presupposed strategy $x\_j$ and we think of one player $i$ deviating from... | 10 | https://mathoverflow.net/users/5963 | 49318 | 31,050 |
https://mathoverflow.net/questions/49315 | 28 | In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}\*\mathbb{Z}/2$ because clearly every element can be written as a product of $b$'s (only to the power 1) and powers of $a$.... | https://mathoverflow.net/users/6646 | What group is $\langle a,b \,| \, a^2=b^2 \rangle$? | Setting $c:=b^{-1}$ one obtains the presentation
$$G= \langle a,c \, | \, a^2c^2=1 \rangle,$$
which is the fundamental group of the Klein bottle.
It is well known that another presentation of such a group is
$$G= \langle x,y \,|\, x^{-1}yx=y^{-1} \rangle,$$
and this allows one to write $G$ as a semi-direct product o... | 63 | https://mathoverflow.net/users/7460 | 49320 | 31,052 |
https://mathoverflow.net/questions/49278 | 22 | For a connected reductive algebraic group $G$ over a field $k$, other than the \'etale fundamental group of $G$ (regarded just as a scheme), there seems to be another notion, usually called the algebraic fundamental group of $G.$ I am not sure of its definition, but I guess (at least when $G$ is split) it might be some... | https://mathoverflow.net/users/370 | The algebraic fundamental group of a reductive algebraic group | At Jim's request, here's an expanded version of my comments above. I will have to use some facts from the topological theory of complex algebraic varieties, but out of stubbornness I will not use any such facts which are part of the theory of Lie groups (the maximal compact subgroup, facts specific to complex semisimpl... | 51 | https://mathoverflow.net/users/3927 | 49342 | 31,064 |
https://mathoverflow.net/questions/49056 | 7 | Hello,
is the problem of pattern recognition (for a given sequence of n numbers, find the shortest Turing machine with an alphabet of 42 elements that will output these n numbers in, say, 5\*n^3 time) an NP-complete problem?
To me, it seems practically impossible that the problem is in P: If you had the level of un... | https://mathoverflow.net/users/11506 | Is pattern recognition NP-complete? | The key phrase you are looking for is "**resource-bounded Kolmogorov complexity**". [This paper](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.9495) by Allender, et. al. may be a good starting point. Also, [this PhD thesis](http://www.research.rutgers.edu/~troyjlee/thesis.html) might provide some helpful ... | 5 | https://mathoverflow.net/users/9840 | 49344 | 31,066 |
https://mathoverflow.net/questions/49255 | 4 | Given a Vandermonde matrix
$
V=
\begin{bmatrix}
1 & 1 & 1 & \ldots & 1 \\\\
x\_1 & x\_2 & x\_3 & \ldots & x\_n \\\\
x\_1^2 & x\_2^2 & x\_3^2 & \ldots & x\_n^2 \\\\
\vdots & \vdots & \vdots & \ddots & \vdots \\\\
x\_1^{m-1} & x\_2^{m-1} & x\_3^{m-1} & \ldots & x\_n^{m-1}
\end{bmatrix},
$
when $m=n-1$, $x\_i \neq x\_j$... | https://mathoverflow.net/users/10705 | How to determine the kernel of a Vandermonde matrix? | The answer is pretty much given by darij but it is nice enough (in final form) to spell out a bit further. The short story is that for $n=4$ one vector in (right) kernel is the column vector
$[\frac{1}{(x\_1-x\_2)(x\_1-x\_3)(x\_1-x\_4)},\frac{1}{(x\_2-x\_1)(x\_2-x\_3)(x\_2-x\_4)},\frac{1}{(x\_3-x\_1)(x\_3-x\_2)(x\_3... | 11 | https://mathoverflow.net/users/8008 | 49346 | 31,067 |
https://mathoverflow.net/questions/49336 | 6 | Fix a hyperkähler manifold $X$ and an identification of $S^2$ with the hyperkähler sphere of $X$. Now consider the twistor space $T := S^2\times X$ equipped with the tautological complex structure. For each $x\in X$, we have a holomorphic map $u\_x:S^2\to T$ defined by $u\_x(\theta):=(\theta,x)$.
**Question:** Is eve... | https://mathoverflow.net/users/10934 | Holomorphic spheres in hyperkähler twistor spaces | I think the preprint [arXiv:1006.0440](https://arxiv.org/abs/1006.0440) of Jardim and Verbitsky will answer your question. In short the answer is no, since if $\dim\_C X = n$ then the deformation space of sections has dimension $\dim H^0(S^2,N\_{S^2/S^2\times X}) = \dim H^0(P^1,O\_{P^1}(1)^n) = 2n$ which is twice the d... | 4 | https://mathoverflow.net/users/4428 | 49347 | 31,068 |
https://mathoverflow.net/questions/49349 | 10 | The only examples I have encountered of infinite $p$-groups with trivial center employ non-elementary methods in their construction. For instance, Example 9.2.5 of Scott's *Group Theory* is a perfectly satisfactory example, but it requires the wreath product (which, though an invaluable group-theoretic tool, is not wha... | https://mathoverflow.net/users/11445 | Simple(st) example of an infinite $p$-group with trivial center | Here is a matrix example, hopefully correct and also hopefully sufficiently simple:
Consider the group $U$ of $\infty\times \infty$ upper unipotent matrices with entries in
$\mathbb F\_p$, with all but finitely many entries equal to zero;
so they have the form
$$\begin{pmatrix} 1 & a\_{1 2} & a\_{1 3} & \cdots \\\ 0... | 18 | https://mathoverflow.net/users/2874 | 49352 | 31,069 |
https://mathoverflow.net/questions/49365 | 7 | Generate $S\_n$ by transpositions $s\_i$ of (i) and (i+1). Both $S\_3$ and $S\_4$ have single elements of maximal word norm associated with this presentation. In fact, the Cayley graph of $S\_3$ can be seen as a tiling of $S^1$, and the Cayley graph of $S\_4$ a tiling of $S^2$. The element of maximal length is then ant... | https://mathoverflow.net/users/11557 | Does every symmetric group S_n have a single element of maximal word norm? | It is amazing how a fact that I was taught in a middle school can be proved using big theories where I don't understand half of the words. Let me add a straightforward proof (for $S\_n$ and only $S\_n$).
For a permutation $\sigma:\{1,\dots,n\}\to\{1,\dots,n\}$, let $\lambda(\sigma)$ denote the number of inversions in... | 33 | https://mathoverflow.net/users/4354 | 49372 | 31,081 |
https://mathoverflow.net/questions/49357 | 34 | In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G\_2$. I am wondering if people could expand on the geometry related to the exceptional Lie Groups. I am not precisely sure what I am looking for, but ostensibly there should be answers forth coming from other who... | https://mathoverflow.net/users/3901 | $G_2$ and Geometry | I promised Sean a detailed answer, so here it is.
As José has already mentioned, it is only $G\_2$ (of the five exceptional Lie groups) which can arise as the holonomy group of a Riemannian manifold. Berger's classification in the 1950's could not rule it out, and neither could he rule out the Lie group $\mathrm{Spin... | 84 | https://mathoverflow.net/users/6871 | 49389 | 31,092 |
https://mathoverflow.net/questions/49395 | 65 | I asked myself the following question while preparing a course on power series for 2nd year students. Let $F$ be the set of power series with convergence radius equal to $1$. What subsets $S$ of the unit circle $C$ can be realised as
$$
S:=\{x \in C: f\text{ diverges in }x\}
$$
for $f \in F$? Any finite subset (and po... | https://mathoverflow.net/users/11563 | Behaviour of power series on their circle of convergence | [**Edit (Jan.12/12):** I've recently been made aware of additional results. I have added an update at the end.]
Hi Piotr,
As far as I know, the question is still open, and not much seems known beyond the classic results of George Piranian and Fritz Herzog. They are contained in two joint papers, available at the pu... | 68 | https://mathoverflow.net/users/6085 | 49411 | 31,103 |
https://mathoverflow.net/questions/48899 | 7 | Let $A$ be a non-zero commutative ring with unit, $I$ a infinite set.
>
> Can $\prod\_{i\in I}A$ be free as an $A$-module?
>
>
>
I found when $A$ is a field or is isomophic to $\mathbb{Z}/m\mathbb{Z}$, then it is free.
But even when $A=\mathbb{Z}$, it is not free. (Baer Specker group)
It seems $\prod\_{i\i... | https://mathoverflow.net/users/11218 | Can the I-fold direct product be free? | If $A$ is a noetherian domain and not a field then the infinite product $M=A\times A\times \dots$ is not free. Suppose there is a basis. For $x\in M$ define its support to be the finite set of basis elements for which the coefficient is not zero. Note that if the supports of $x$ and $y$ are disjoint then their union is... | 11 | https://mathoverflow.net/users/6666 | 49422 | 31,110 |
https://mathoverflow.net/questions/49425 | 2 | Given a set with a known cardinality, least upper and greatest lower bound; how can I calculate the maximum possible standard deviation for any set of values within the set.
As an example:
Given a set {1,50}, with mean of 25.5, the std. dev. for both members at 34.65.
This std. dev. is the possible for a set with ... | https://mathoverflow.net/users/1436 | Calculate largest possible standard deviation for a set | The sample deviation is maximized when half of the observations are at each extreme. If you want a formula for the maximum standard deviation as a function of the interval endpoints and the number of samples, you will probably want to divide the formula up depending on whether the number of samples is even or odd.
| 1 | https://mathoverflow.net/users/11570 | 49427 | 31,113 |
https://mathoverflow.net/questions/49415 | 85 | The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's *Principles of Mathematical Analysis*, following Theorem 3.29, he writes:
>
> One might thus be led to conjecture that there is a limiting situation of some sort, a “... | https://mathoverflow.net/users/3106 | Nonexistence of boundary between convergent and divergent series? | A rather detailed discussion of the subject can be found in Knopp's [*Theory and Application of Infinite Series*](https://books.google.com/books?id=qLPxxeBzp84C "zbMATH review at https://zbmath.org/?q=an:0042.29203") (see § 41, pp. 298–305). He mentions that the idea of a possible boundary between convergent and diverg... | 41 | https://mathoverflow.net/users/5371 | 49429 | 31,115 |
https://mathoverflow.net/questions/49391 | 3 | Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $B$ is a discrete valuation ring such that $B[\pi^{-1}]=K(t)$.
Can the residue field of $B$ be an algebraic extension... | https://mathoverflow.net/users/345 | How exotic can DVRs be in the ring of rational functions over a local field? | I will suppose $k$ is **perfect** for simplicity.
**Statement**: There exists $B$ with algebraic residue extension $k\_B/k$ if and only if $k$ has infinite index in its algebraic closure. And this implies that $[k\_B : k]=+\infty$.
First we prove the following facts:
(1) Let $F/K$ be an extension of discrete va... | 8 | https://mathoverflow.net/users/3485 | 49452 | 31,134 |
https://mathoverflow.net/questions/49451 | 8 | Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate symmetric forms on $\mathfrak{g}$:
1. Standard Killing form: $K(X,Y)=tr(ad\_X\circ ad\_Y)$
2. "Killing-like" form assoc... | https://mathoverflow.net/users/11521 | Killing form vs its counterpart in a given represenation | They are proportional if $g$ is simple. The form $K\_\phi$ defines a homomorphism from the adjoint to the coadjoint representation. If the adjoint representation is irreducible, i.e. $g$ is simple, you know all such homomorphisms are proportional by Schur's lemma.
| 5 | https://mathoverflow.net/users/5301 | 49455 | 31,137 |
https://mathoverflow.net/questions/49435 | 2 | For the Hopf algebra $SL\_q(N)$ it is clear that the kernel of the counit contains the ideal generated by the elements $(u^i\_i-1)$ and $u^i\_j$, for $i \neq j$. However, I cannot seem to arrive at at proof that it is in fact generated by these elements. Does anyone how to do this?
| https://mathoverflow.net/users/2612 | Generators of the Augmentation Ideal (Counit Kernel) | Just observe that the quotient by the ideal generated by these elements is at most 1-dimensional.
| 3 | https://mathoverflow.net/users/5301 | 49456 | 31,138 |
https://mathoverflow.net/questions/49400 | 9 | Is it true that in any successive (natural) $2p\_n$ numbers there are at least three numbers that are not divisible by any prime less (not equal) than $p\_n$? Here, $p\_n$ denotes the $n$-th prime number.
For
example in any six successive numbers there are at least 3 numbers that are not divisible by 2,in any 10 suc... | https://mathoverflow.net/users/14726 | question in prime numbers |
>
> **Late update** A better question might be; "What is the [Largest number of consecutive integers such that each is divisible by a prime $\le p\_n$](http://oeis.org/A058989) ? Computing a few values and checking the handy OEIS yields the link above. Matching the values with the appropriate primes shows that one ca... | 12 | https://mathoverflow.net/users/8008 | 49463 | 31,142 |
https://mathoverflow.net/questions/49461 | 7 | Is there a reference or a very short argument proving the following statement?
Let $C$ be a set consisting of $r$ points in the real projective space $\mathbb RP ^k$
with its usual round metric. Assume that the distances between all pairs of points in $C$ are the same. Assume further that $r>k+1$. Then $r$ must be eq... | https://mathoverflow.net/users/10086 | Regular simplex in projective space | Equidistant points in projective space correspond to equiangular lines in Euclidean space. The maximum number of equiangular lines in $\mathbb R^n$, $f(n)$ is a function which might not be as nice as you think, some computed values are in [OEIS](http://oeis.org/A002853). For example, your claim fails even in the case $... | 17 | https://mathoverflow.net/users/2384 | 49470 | 31,148 |
https://mathoverflow.net/questions/49418 | 10 | A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a **continuous** real function is that it is the union of a $G\_\delta$ set and a $G\_{\delta \sigma}$ set of zero measure.
On the other hand it is not hard to see that the... | https://mathoverflow.net/users/8584 | Nondifferentiability set of an arbitrary real function | Apparently, continuity is not essential.
According to A. Brudno, *Continuity and differentiability* (Russian), Rec. Math [Mat. Sbornik], N.S. 13 (**55**), (1943), 119–134 ([MathSciNet review here](http://www.ams.org/mathscinet-getitem?mr=12321), [online article here](http://mi.mathnet.ru/eng/msb6175)), the set of non... | 6 | https://mathoverflow.net/users/11081 | 49487 | 31,156 |
https://mathoverflow.net/questions/49486 | 9 | ### Convention:
Before I start, please note that it is not going to be sufficient to assume that the topology is subcanonical or that the site $C$ has finite limits, since the application I have in mind does not satisfy either of these two criteria. Therefore, we take our definition of a topology to be the one in ter... | https://mathoverflow.net/users/1353 | The single-plus construction is not the left adjoint of the inclusion of separated presheaves? | If $(-)^+$ were left adjoint to $\iota$, we would have $(-)^+\circ\iota=id.$, but this is impossible since $(-)^+$ applied to a separated presheaf gives its sheafification.
| 18 | https://mathoverflow.net/users/5513 | 49489 | 31,158 |
https://mathoverflow.net/questions/49485 | 1 | What ie the meaning of the statement : Modular Lambda Function is such that,over any point $\tau$ in the upper half plane, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve $\mathbb{C}/\langle 1, \tau \rangle$
| https://mathoverflow.net/users/11583 | Modular Lambda Function as a Cross ratio | The elliptic curve $\mathbb{C}/\langle 1, \tau \rangle$ is an abelian group with a "-1" automorphism, given by sending each point $z$ to $-z$. If we take the quotient by this automorphism, we get a projective line. Another way to say that is there is an action of a group of order two on the elliptic curve that yields a... | 2 | https://mathoverflow.net/users/121 | 49498 | 31,162 |
https://mathoverflow.net/questions/49459 | 6 | I asked this as subquestion in a comment pursuant to my Banach-Tarski
question. I think it is worth promoting here to a question in its own right.
Consider these two matrices over ${\Bbb R}[[\epsilon]]$:
$$A = \left[ \begin{array}{ccc}
\cos(\epsilon) & \sin(\epsilon) & 0 \\
-\sin(\epsilon) & \cos(\epsilon) & 0 \\
0 ... | https://mathoverflow.net/users/10909 | Words in two infinitismal rotations | This question is related (loosely) to the theory of discrete groups generated by small elements, which was developed and used for various purposes by Margulis and others. If $A$ and $B$ are elements of a Lie group, then in exponential coordinates around the identity, multiplication looks like addition to first order, a... | 7 | https://mathoverflow.net/users/9062 | 49501 | 31,163 |
https://mathoverflow.net/questions/49494 | 5 | The [generalized Erdős–Heilbronn (GEH) theorem](http://en.wikipedia.org/wiki/Restricted_sumset), which is proved by da Silva and Hamidoune in 1994, states that:
**Theorem.** If p is a prime and $X$ is a subset of $\mathbb{Z}\_p$, then $|\hat{k}X| \geq \min \lbrace k|X|-k^2+1 , p \rbrace$ for $\hat{k}X = \lbrace x\_1+... | https://mathoverflow.net/users/4248 | Is the generalized Erdős–Heilbronn problem true for finite cyclic groups? | Arithmetic progressions usually give small sumsets so...How about $X=\lbrace 0,3,6,9,12 \rbrace \subset \mathbb{Z}\_{15}$? Then for $k=2,3,4$ one has $\hat{k}X=X$. That is ok for $k=4$ but not for $k=2,3$. (Doesn't that contradict what you said about $k=2$?)
In fact $X=\lbrace 0,d,2d,\cdots,nd\rbrace \subset \mathbb{... | 5 | https://mathoverflow.net/users/8008 | 49502 | 31,164 |
https://mathoverflow.net/questions/49507 | 11 |
>
> Suppose $A$ is a non-zero ring (say commutative unital) and $I$ is an infinite set. Can it happen that there is an isomorphism of $A$-modules $\bigoplus\_{i\in I}A\cong \prod\_{i\in I}A$?
>
>
>
The obvious morphism $\bigoplus\_{i\in I}A\to \prod\_{i\in I}A$ is obviously not an isomorphism, but could there be... | https://mathoverflow.net/users/1 | Can ⨁_I A be isomorphic to ∏_I A for infinite I? | The answer to the first question is no. Here is a proof.
Let $A$ be a commutative ring and $\mathfrak{m}$ be a maximal ideal of $A$. Let $k=A/\mathfrak{m}$. Suppose we have an isomorphism of $A$-modules $A^I \cong A^{(I)}$, where $A^{(I)} := \bigoplus\_{i \in I} A$. Tensoring with $k$ over $A$ yields an isomorphism o... | 9 | https://mathoverflow.net/users/6506 | 49514 | 31,172 |
https://mathoverflow.net/questions/49516 | 15 | I need to compute the Voronoi diagram of a set of points in $R^n$.
I'm quite unschooled on the topic, could someone point me to the right references so that I can
a) understand the theory behind it;
b) implement an actual algorithm to compute it.
>
> P.S. Googling I found out I need to know about euclidean grap... | https://mathoverflow.net/users/11588 | $n$-dimensional Voronoi diagram | Often Voronoi diagrams are constructed from their duals, Delaunay triangulations.
The Delaunay triangulation of a point set in $d$ dimensions can be obtained from the convex hull
of a lifting of the points into $d+1$ dimensions. This relationship is explained in many sources,
including [*The Handbook of Discrete and Co... | 18 | https://mathoverflow.net/users/6094 | 49521 | 31,176 |
https://mathoverflow.net/questions/49394 | 6 | I hope to know the classification of real forms of complex simple Lie algebras of types $A$, $D$, $E$ up to inner automorphisms.
Let $\mathfrak{g}\_1$ and $\mathfrak{g}\_2$ be real forms of a complex simple Lie algebra $\mathfrak{g}$.
We say that they are equivalent if there is an isomorphism $\mathfrak{g}\_1 \to \ma... | https://mathoverflow.net/users/11562 | Classification of real forms up to inner automorphisms | The one case (for $\frak g$ simple) where the two definitions of real forms (up to $Aut(\frak g)$ versus $Int(\frak g)$) don't agree is the following.
There is a real form of $\frak g=\frak s\mathfrak o(2n,\mathbb C)$ denoted
$\frak s\frak o^\*(2n)$.
For $n\ge4$ even $\frak g$ has two subalgebras
isomorphic to $... | 3 | https://mathoverflow.net/users/6030 | 49525 | 31,178 |
https://mathoverflow.net/questions/49524 | 1 | Free functors are left adjoint to forgetful one. If given two free functors F:Set->A,G:Set->B constructing objects of type A & B respectively, is there a canonical free functor F\*G that constructs objects that are both of type A & B?
This is probably too simple minded, but since Cat is bicomplete, I considered the p... | https://mathoverflow.net/users/6408 | Is there a canonical 'composition' of Free Functors? | Under the most immediate reading of "both of type A and B", where no sort of compatibility or relationship between the A and B structures is assumed, the answer is no.
A counterexample is where A is the theory of sup-lattices and B is the theory of sets equipped with an unary operator (often called "successor"). If ... | 10 | https://mathoverflow.net/users/2926 | 49527 | 31,179 |
https://mathoverflow.net/questions/49279 | 3 | Let $\mu:X'\rightarrow X$ be a birational morphism of normal complex projective varieties.
Consider the two ideal sheaves $I\_1= \mu\_\*\mathcal{O}\_{X'}(-\sum d(E)E)$, $I\_2=\mu\_\*\mathcal{O}\_{X'}(-\sum(d(E)+1)E)$, where the $d(E)$'s are non negative integers and the $E$'s are prime divisors.
Suppose $x\in X$ i... | https://mathoverflow.net/users/6430 | About direct image of ideal sheaves | The answer is no.
Consider $\mu=\mu\_z \circ \mu\_y \circ \mu\_x$ the blow up of a smooth surface at three points $x$, $y$, $z$, as follows: $x\in X$ is arbitrary, $y\in E\_x:=\mu\_x^{-1}(x)$, where $\mu\_x$ is the blowup of $X$ centered at $x$, and $z$ is the "satellite" point that appears after blowing up $y$, ie, ... | 4 | https://mathoverflow.net/users/1939 | 49534 | 31,181 |
https://mathoverflow.net/questions/49426 | 64 | The usual category of measure spaces consists of objects $(X, \mathcal{B}\_X, \mu\_X)$, where $X$ is a space, $\mathcal{B}\_X$ is a $\sigma$-algebra on $X$, and $\mu\_X$ is a measure on $X$, and measure preserving morphisms $\phi \colon (X, \mathcal{B}\_X, \mu\_X) \to (Y, \mathcal{B}\_Y, \mu\_Y)$ such that $\phi\_\ast ... | https://mathoverflow.net/users/11568 | Is there a category structure one can place on measure spaces so that category-theoretic products exist? | To clarify Chris Heunen's answer, let me point out that most notions of measure theory
have analogs in the category of smooth manifolds.
For example, the analog of a measure space (X,M,μ), where X is a set, M is a σ-algebra of measurable subsets of X, and μ is a measure on (X,M),
is a smooth manifold X equipped with a ... | 73 | https://mathoverflow.net/users/402 | 49542 | 31,185 |
https://mathoverflow.net/questions/49515 | 1 | I am not an expert in combinatorics, but I need to count (or to approximate) the number of antichains of a poset.
The idea relies on approximating this number by embedding my poset into another one, that I call "rectangular" but actually I don't know if there is already a standard definition.
Let's say that my pose... | https://mathoverflow.net/users/6882 | On the number of antichains of a poset | You appear to switch the usage of $d$ and $a$ when you identify your poset with a subset of the other; in what follows, I am using $d$ and $a$ as you defined them, i.e., $d$ is the maximum chain length and $a$ is the maximum antichain size.
Elaborating on Dave Pritchard's point 1: by Dilworth's Theorem, there exist $... | 3 | https://mathoverflow.net/users/4658 | 49550 | 31,191 |
https://mathoverflow.net/questions/49548 | 34 | Consider an algebraic vector bundle $E$ on a scheme $X$. By definition there is an open cover of $X$ consisting of open subsets on which $E$ is trivial and if $X$ is quasi-compact, a finite cover suffices. The question then is simply: what is the minimum number of open subsets for a cover which trivializes $E$ ? Now th... | https://mathoverflow.net/users/450 | An algebraic vector bundle is trivialized by open sets. How many does one need? | This is true if we assume that the vector bundles has constant rank (it is clearly false if we allow vector bundles to have different ranks at different points). Let $U\_1$ be an open dense subset of $X$ over which $E$ is trivial, and let $H\_1$ be a hypersurface containing the complement of $U\_1$. Then $E$ is trivial... | 31 | https://mathoverflow.net/users/4790 | 49560 | 31,196 |
https://mathoverflow.net/questions/49526 | 4 | I'm teaching an elementary class about fundamental groups and covering spaces. It was very useful to use "fool's covering spaces" of a space $X$, defined as
functors $\Pi\_1(X)\to Sets$, where $\Pi\_1(X)$ is the fundamental groupoids of $X$. In a more "covering space way", a fool's covering space can be described as a ... | https://mathoverflow.net/users/9390 | Terminology question on covering spaces | "Fool's covering spaces" are very close to *overlays* of R. H. Fox (see [this](http://www.springerlink.com/content/hk52316r65043102/) paper in the first place and also [this one](http://matwbn.icm.edu.pl/ksiazki/fm/fm74/fm7416.pdf)), which I think are still better: they retain all nice properties of "fool's covering sp... | 6 | https://mathoverflow.net/users/10819 | 49570 | 31,200 |
https://mathoverflow.net/questions/49551 | 33 | This question is motivated by the question [link text](https://mathoverflow.net/questions/49507/can-i-a-be-isomorphic-to-i-a-for-infinite-i), which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is never isomorphic to its dual. Mo... | https://mathoverflow.net/users/6506 | Dimension of infinite product of vector spaces | The answer to both questions is yes.
As a preliminary, let's prove that for any infinite-dimensional vector space $V$, that
* **Lemma:** $|V| = |k| \cdot \dim V$
where $|X|$ denotes the cardinality of a set $X$.
Proof: Since $|k| \leq |V|$ and $\dim V \leq |V|$, the inequality
$$|k| \cdot \dim V \leq |V|^2 = ... | 32 | https://mathoverflow.net/users/2926 | 49572 | 31,201 |
https://mathoverflow.net/questions/28109 | 2 | Let $X$ be a compact Kaehler manifold.
What is a good, possibly algebraic-geometric, way to think to Nakano semipositivity of holomorphic vector bundles on $X$?
Is the trivial line bundle $\mathcal{O}\_X$ Nakano semi-positive as a vector bundle?
| https://mathoverflow.net/users/6430 | Nakano semipositivity | I would say more properly that nowadays it is not known any satisfactory algebraic description or characterization of the concept of Nakano's positivity for a hermitian vector bundle.
I would like also to add some precisions to Sándor's answer.
First, the positivity in the sense of Nakano is a good notion to obtain... | 5 | https://mathoverflow.net/users/9871 | 49574 | 31,202 |
https://mathoverflow.net/questions/49566 | 2 | I have seen two proofs of the Sobolev embedding theorem. One uses the Hardy-Littlewood-Sobolev inequality
$
\displaystyle \|f\|\_{L^q({\bf R}^d)} \leq C\_{p,q,d} \|f\|\_{W^{1,p}({\bf R}^d)}.
$
One uses (for $p>1$) the Hardy-Littlewood-Sobolev theorem on fractional integration and the Gagliardo-Nirenberg inequality ... | https://mathoverflow.net/users/11604 | A question about the proofs of the Sobolev embedding theorem. | The classical Gagliardo-Nirenberg proof covers the limiting cases
$$\|f\|\_{L^{\frac{n}{n-1}}}\le C\|\nabla f\| \_{L^1}$$
and, for $f$ vanishing at infinity,
$$\|f\| \_{L^\infty}\le C\|f\| \_{W^{n,1}}$$
which can not be easily recovered using the HLS inequality. On the other hand, HLS allows for a unified and simple pr... | 6 | https://mathoverflow.net/users/7294 | 49578 | 31,204 |
https://mathoverflow.net/questions/48760 | 7 | For a locally compact (Hausdorff) abelian group $G$ we have following theorem (see e.g. Folland):
"For every (strongly continuous) unitary representation $(\pi,\mathcal{H\_{\pi}})$ of $G$, there exists a unique regular $\mathcal{H}\_{\pi}$-[projection-valued measure](http://en.wikipedia.org/wiki/Projection-valued_mea... | https://mathoverflow.net/users/7392 | Decomposing an arbitrary unitary representation of a connected nilpotent Lie group in terms of its irreps | First some bad news : such a decomposition only exists for groups which are said to be of Type I (some notion coming from the theory of von Neumann algebras). There are examples of topological groups which are not of Type I, and have representations which can be decomposed in two different way (even with disjoint suppo... | 3 | https://mathoverflow.net/users/9962 | 49579 | 31,205 |
https://mathoverflow.net/questions/19936 | 13 | I am thinking about the precise formulation of the Lefschetz duality for the relative cohomology. If I understand [this Wikipedia article](http://en.wikipedia.org/wiki/Lefschetz_duality) correctly, there is an isomorphism between $H^k(M, \partial M)$ and $H\_{n-k}(M)$ and hence (I suppose) a non-degenerate pairing $H^k... | https://mathoverflow.net/users/3909 | Pairing used in Lefschetz duality | Yes, your formula is right. For the intuitive understanding just compute it for 1- and 2- dimensional half-spaces.
See Bott & Tu, Differential forms in Algebraic topology, $\S 5$, Poincaré duality.
I give only sketch of proof for your question.
1. First of all you need pairing between $H\_c^k(M, \partial M)$ and ... | 5 | https://mathoverflow.net/users/4298 | 49583 | 31,208 |
https://mathoverflow.net/questions/49573 | 1 | My question really is:
if $e^{2\pi i\* g(\theta)}$ is an algebraic function in the variable $e^{2 \pi i \theta}$, what restrictions can we put on g?
My first guess is to say that g is the map that sends everything to zero, or $g(\theta)=n\theta +c$, in which case $e^{2\pi i\* g(\theta)}=1$ or $C\*(e^{2 \pi i \thet... | https://mathoverflow.net/users/4249 | Looking for Direction on algebraic and transcendental functions | (You should change the tag, this has nothing to do with functional analysis)
By Ax's theorem giving a function field analogue of Schanuel's conjecture, if $g(\theta)$ is an algebraic function of $\theta$ such that $e^{2\pi i \theta}$ and $e^{2 \pi i g(\theta)}$ are algebraically dependent, then $\theta$ and $g(\theta... | 2 | https://mathoverflow.net/users/2290 | 49584 | 31,209 |
https://mathoverflow.net/questions/49609 | 1 | Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of the form $t=ah$ for $a\in \mathbb C$.
>
> Can we conclude that $\mathcal F$ is flat?
>
>
>
For coherent sheaves... | https://mathoverflow.net/users/66 | Flatness on the formal plane from flatness on lines through the origin? | On $X=\mathrm{Spec}\,\mathbb{C}[[t,h]]$, choose an irreducible curve of degree $\geq2$ (e.g. $t^2=h$) and let $\eta$ be its generic point, $j:\eta\to X$ the inclusion. Then $\mathcal{F}:=j\_\*\mathcal{O}\_\eta$ is quasicoherent and zero on every line through the origin, but not flat. (In terms of $\mathbb{C}[[t,h]]$-mo... | 8 | https://mathoverflow.net/users/7666 | 49613 | 31,225 |
https://mathoverflow.net/questions/49622 | 3 | A real polynomial $f(x\_1,\ldots, x\_n)$ in several variables is a *sum of squares* if there are polynomials $g\_1,\ldots, g\_k$ such that $f=g\_1^2+\cdots +g\_k^2$.
Fix a positive number $d>0$. The collection of real polynomials of degree d ${\mathbb{R}}[x\_1,\ldots, x\_n]\_d$ has the structure of a finite-dimension... | https://mathoverflow.net/users/nan | Is the set of polynomial sum of squares closed under limits? | The cone of sums of squares $\Sigma^2 \subset \mathbb R[x\_1,\dots,x\_n]$ is closed in the finest locally convex topology. This is equivalent to the assertion that the intersection of this cone with the space of polynomials up to degree $d$ is closed in the usual euclidean topology for every $d$.
The argument goes a... | 6 | https://mathoverflow.net/users/8176 | 49629 | 31,231 |
https://mathoverflow.net/questions/49595 | 1 | In my [recent question](https://mathoverflow.net/questions/49328/double-duality-for-geometrically-defined-graph-imbeddings) I asked about a proof for the fact that the dual of a dual graph imbedding is equal to the original graph. Thinking about this a little more leads me to wonder why graph imbeddings are defined usi... | https://mathoverflow.net/users/960 | Why are graph imbeddings defined the way they are? | I can think of a couple of other reasons. The first is *algorithms*. Many algorithmic problems become easier if we are told that the input graph is embeddable on a surface. As a trivial example: the problem of deciding if a graph is 4-colourable is NP-hard in general, but pretty damn easy when restricted to the class o... | 7 | https://mathoverflow.net/users/2233 | 49631 | 31,232 |
https://mathoverflow.net/questions/49561 | 3 | Edit 1: I think that the question was not stated clearly enough so modified it a little.
Edit 2:
I thought over the physics that lies behind this question which led me to reformulation of the original problem. Orbit itself is non physically significant. What really counts is its image in projective space!
Edit 3:
I... | https://mathoverflow.net/users/11521 | Highest weight orbit characterization (reformulated and extended) | Let $H$ be the isotropy group of the highest weight ray $\pi(v\_\lambda)$ in $G\_0$ and
$P$ the highest weight ray isotropy group in $G$. It is easy to see that $H$ is the
centralizer of the torus generated by the coroots corresponding to the nonvanishing components of $\lambda$ in the weight basis, which is a conseque... | 2 | https://mathoverflow.net/users/1059 | 49635 | 31,235 |
https://mathoverflow.net/questions/49619 | 6 | Hello,
let $U$ be the assertion "The union of $\aleph\_{1}$ null sets of reals is null", i.e.
$U$ = Given any $\omega\_{1}$-sequence of null sets $X\_{\alpha}$, for $\alpha<\omega\_{1}$, then $\bigcup\_{\alpha <\omega\_{1}} X\_{\alpha}$ is null.
$U$ is known to be independent of ZFC.
It clearly does not hold if ... | https://mathoverflow.net/users/11618 | Omega_{1} unions of null sets: Martin's Axiom | These questions and many other similar questions are
intensely studied in the field known as cardinal
characteristics of the continuum. Some ideas are mentioned
in [this MO
answer](https://mathoverflow.net/questions/8972#9027) (and
also in [this MO answer](https://mathoverflow.net/questions/29624/how-many-orders-of-inf... | 9 | https://mathoverflow.net/users/1946 | 49639 | 31,237 |
https://mathoverflow.net/questions/43474 | 6 | This is a question that stemmed from [fooling around](http://www.cs.mcgill.ca/~akazna/kaznatcheev20100509.pdf) with unitary t-designs.
Let
\begin{equation}
\mathbb{V} = \mathrm{span} \{\; U^{\otimes t}\; |\; U \in \mathrm{U}(d)\}
\end{equation}
Where $\mathrm{U}(d)$ is the unitary group acting on $\mathbb{C}^d$... | https://mathoverflow.net/users/8239 | Symmetric subspace of linear operators | The proof below is mostly by [Peter Scholze](https://mathoverflow.net/users/6074/peter-scholze), so no point in voting it up.
1° I claim that $\dim \mathbb V=\binom{d^2+t-1}{t-1}$. In order to prove this, I will show that the subset $\mathrm{U}\_d$ (this is what you call $\mathrm{U}\left(d\right)$ and is defined as t... | 4 | https://mathoverflow.net/users/2530 | 49650 | 31,242 |
https://mathoverflow.net/questions/49633 | 3 | I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has to choose (at least if one is not willing to deal with forms of odd parity).
While I was wondering how to extend this ... | https://mathoverflow.net/users/1841 | How to define the orientation of a vector space over an arbitrary field? | An orientation of the $n$-dimensional real vector space $V$ is an equivalence class of generators of the $1$-dimensional vector space $det(V)=\Lambda^n(V)$ under the relation $\omega\sim c\omega$, $c>0$.
A basis-free description of the usual Hodge star for a real vector space with positive inner product is exactly as... | 11 | https://mathoverflow.net/users/6666 | 49652 | 31,243 |
https://mathoverflow.net/questions/49644 | 12 | Let $C$ be a smooth projective curve and $E$ a vector bundle of rank $r$ on $C$. We say that $E$ is nef/ample if $\mathcal{O}\_{\mathbb{P}(E)}(1)$ is so. Equivalently (see Hartshorne's papers on 'Ample vector bundles' and 'Ample vector bundles on curves'), $E$ is ample if and only if for any coherent $F$, $S^m(E)\otime... | https://mathoverflow.net/users/386 | Ample vector bundles on curves | Here's a partial answer. Suppose we're in characteristic 0 (Fujita would be assuming this),
and that $rank(E)=2$. By cor 7.6 of Hartshorne's ample vector bundles paper, it suffices
to check that $deg(E)>0$ and $deg(L)>0$ for ay quotient line bundle.
From Riemann-Roch as in Piotr's comment, we get
$$deg(E) + rank(E)(1-g... | 8 | https://mathoverflow.net/users/4144 | 49656 | 31,246 |
https://mathoverflow.net/questions/49532 | 3 | Let $A$ be a $n \times n$ matrix with non-negative entries $a\_{ij}$, where $a\_{ij}$ is the entry in the $i^{th}$ row and $j^{th}$ column. Assume $\sum\_{1 \leq j \leq n} a\_{ij} \leq 1$ for all $1 \leq i \leq n$. Also assume $a\_{ii} = 0$ for all $1 \leq i \leq n$.
I want to partition the index set $I = \{1, 2 \ld... | https://mathoverflow.net/users/5873 | Partitioning a matrix with bounded row sums | Ok, I think there are examples where $\Omega(\log n)$ colors are needed.
Here’s an example, let $a\_{ij} = \frac{1}{i}$ for $j < i$ and $a\_{ij} = \frac{1}{j^2}$ for $j > i$. Then $\sum\_{j} a\_{ij} = \frac{i-1}{i} + \sum\_{j > i} \frac{1}{j^2} = O(1)$. Of course, the bound is $O(1)$ instead of $1$, but that can be ... | 0 | https://mathoverflow.net/users/5873 | 49657 | 31,247 |
https://mathoverflow.net/questions/49659 | 2 | I'm somewhat confused about the definitions of Betti numbers for Riemannian manifolds. Working with the first Betti number as an example, I have usually taken the definition to be the rank of the homology group $H\_1(M)$, where $M$ is the manifold in question. I'm also aware that through a Hodge-theoretic argument, we ... | https://mathoverflow.net/users/11266 | Betti Numbers (homology vs cohomology) | $H\_1(M,\mathbb{R}) \cong H^1(M, \mathbb{R})$ follows from the universal coefficients theorem for cohomology.
| 7 | https://mathoverflow.net/users/1106 | 49662 | 31,250 |
https://mathoverflow.net/questions/49679 | 3 | I'd like to know whether the following statement is true or not.
Let $T\_1, T\_2\in \mathbb{C}^{n\times n}$ be upper triangular matrices. If there exists a nonsingular matrix $P$ such that $T\_1=PT\_2P^{-1}$, then there is a nonsingular uppper triangular matrix $T$ such that $T\_1=TT\_2T^{-1}$.
| https://mathoverflow.net/users/3818 | a matrix similarity problem. | It is **false** for the following obvious reason. The diagonal elements of a triangular matrix are its eigenvalue. If they are pairwise distinct, the matrix is similar to its diagonal.
Assume now that two upper triangular matrices have the same diagonal elements, pairwise distinct, but not in the same order. Then th... | 10 | https://mathoverflow.net/users/8799 | 49682 | 31,260 |
https://mathoverflow.net/questions/49678 | 5 | I wonder whether there is a reference for the following two things:
The Grothendieck group of B-equivariant semisimple? perverse sheaves on $G/B$ is the Hecke-algebra.
The category of B-equivariant perverse sheaves on $G/B$ is equivalent to those modules of category $\mathcal O$, where the center acts trivial.
| https://mathoverflow.net/users/2837 | Reference for two facts about perverse sheaves on G/B | Both of these are facts which developed gradually over the course of several papers, so it's hard to give a definitive reference.
For the first, all the calculations one would need in order to establish this fact are done in, for example, Springer's [Quelques applications de la cohomologie d’intersection](http://arc... | 8 | https://mathoverflow.net/users/66 | 49683 | 31,261 |
https://mathoverflow.net/questions/49638 | 4 | Let $X$ be a regular integral projective scheme of dimension 1 over a field $k$ (not algebraically closed). Further, assume $X$ satisfies $dim\_kH^0(X,\mathscr{O}\_X)=1$. Let $\bar{X}$ denote the fibered product $X\times\_k\bar{k}$. Then is it true that $\bar{X}$ is integral?
| https://mathoverflow.net/users/11395 | A question on base change | This is an answer to the updated question, and it is positive in far more general situations. By EGA, IV.9.7.7, for any morphism of finite presentation $X\to Y$, the set $E$ of $y\in Y$ such that $X\_y$ is geometrically integral is locally constructible. In your situation, $Y$ is noetherian and the set of the closed po... | 4 | https://mathoverflow.net/users/3485 | 49692 | 31,264 |
https://mathoverflow.net/questions/49690 | 24 | Let $X$ be a topological space. In elementary algebraic topology, the cup product $\phi \cup \psi$ of cochains $\phi \in H^p(X), \psi \in H^q(X)$ is defined on a chain $\sigma \in C\_{p+q}(X)$ by $(\phi \circ \psi)(\sigma) = \phi(\_p\sigma)\psi(\sigma\_q)$ where $p\_\sigma$ and $\sigma\_q$ denote the restriction of $\s... | https://mathoverflow.net/users/344 | What do cohomology operations have to do with the non-existence of commutative cochains over $\mathbb{Z}$? | Via the Dold-Kan correspondence, the category of cosimplicial abelian groups
is equivalent to the category of nonpositively graded chain complexes of abelian groups
(using homological grading conventions). Both of these categories are symmetric monoidal:
chain complexes via the usual tensor product of chain complexes, ... | 27 | https://mathoverflow.net/users/7721 | 49697 | 31,266 |
https://mathoverflow.net/questions/49555 | 7 | * The vertices are all (ordered) $k$-tuples with distinct components from a universe of size $n$.
* Two vertices are adjacent if their Hamming distance is 1 (i.e. if they differ in exactly one component).
Notice that this graph has $\frac{n!}{(n-k)!}$ vertices. It is an $k(n-k)$-regular graph.
Does the graph have a... | https://mathoverflow.net/users/11598 | Is the following graph well known? | They might have a name, I don't know. For the next few lines let us call each a Partial Permutation graph $PP(n,k)$ (assume $k<n$). They may not get as much respect because they are not distance transitive (which the Hamming, Johnson and Knesser Graphs are) or even distance regular (but see below for the special case $... | 5 | https://mathoverflow.net/users/8008 | 49703 | 31,268 |
https://mathoverflow.net/questions/49702 | 6 | Let $(X, \mathcal B, \mu)$ be a "good" measure space, e.g. $\mu$ is a positive Radon measure on a locally compact topological space $X$ with Borel $\sigma$-algebra $\mathcal B$. Let $A\subset X$ such that every measurable subset of $A$ has zero measure. Is it true that there is a zero measure set $B$ such that $A\subse... | https://mathoverflow.net/users/11056 | Is a subset that contains no positive measurable subsets contained in a null measurable set? | You are asking whether every set with inner measure $0$ has measure $0$ with respect to the completion measure. The Lebesgue measure, for example, does not have this property, since the usual Vitali set is non-measurable, but has inner measure $0$.
I claim that a ($\sigma$-finite) measure space has your property if ... | 16 | https://mathoverflow.net/users/1946 | 49704 | 31,269 |
https://mathoverflow.net/questions/49700 | 6 | It is a famous theorem of Roth, which Szemerédi famously generalized, that if a set of natural numbers has positive upper density then it contains arithmetic progressions of length $k$. The famous Green-Tao Theorem generalized this property to the primes. My question is, is there any progress on the 'inverse' problem?
... | https://mathoverflow.net/users/10898 | Inverse Length 3 Arithmetic Progression Problem for sets with positive upper density | Or just take all powers of $3$ and add to them all numbers that are congruent to $1$ modulo $3$.
| 13 | https://mathoverflow.net/users/1131 | 49708 | 31,273 |
https://mathoverflow.net/questions/49658 | 2 | For a positive integer $n$, let $f(n)$ be the maximum value of $\mathrm{LCM}(S)$ among multisets $S$ of positive integers satisfying $\sum\_{i \in S} (i-1) = n$.
What is known about upper bounds for $f(n)$?
The best that I can do is $\log f(n) \leq \sqrt{n/2} \log (4n/3)$ for $n \geq 6$.
This comes by looking at th... | https://mathoverflow.net/users/1046 | Upper bound for lowest common multiple of integers with (almost) fixed sum | Consider Aaron's suggestion (from his comment). If $p\_1,p\_2,\ldots$ are the primes in order, then for $n\geq p\_1+p\_2+\ldots +p\_k+k$, then $f(n)\geq p\_1p\_2\ldots p\_k$. Using standard analytic number theory estimates, the log of this lower bound of $f(n)$ grows like $O(k \log k)$, while $n$ grows like $O(k^2 log ... | 2 | https://mathoverflow.net/users/425 | 49714 | 31,276 |
https://mathoverflow.net/questions/49721 | 38 | I would like to open a discussion about the [Axiom of Symmetry of Freiling](https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry), since I didn't find in MO a dedicated question. I'll first try to summarize it, and the ask a couple of questions.
**DESCRIPTION**
The *Axiom of Symmetry*, was proposed in 1986 ... | https://mathoverflow.net/users/11618 | Axiom of Symmetry, aka Freiling's argument against CH | The point is that violations of the Axiom of Symmetry are
fundamentally connected with non-measurable sets, and counterexample functions $f$ to AS cannot be nice measurable functions.
You have proved the one direction $CH\to \neg AS$, that if
there is a well-order of the reals in order type
$\omega\_1$, then the func... | 31 | https://mathoverflow.net/users/1946 | 49734 | 31,284 |
https://mathoverflow.net/questions/49732 | 17 | The reading of (Hopkins-)Lurie's On the Classification of Topological Field Theories (arXiv:0905.0465) suggests that a stronger version of the cobordism hypothesis should hold; namely, that (under eventually suitable technical assumptions), the inclusion of symmetric monoidal $(\infty,n)$-categories with duals into $(\... | https://mathoverflow.net/users/8320 | Free symmetric monoidal $(\infty,n)$-categories with duals | The existence of a left adjoint follows by formal nonsense. If you have a symmetric monoidal $(\infty,n)$-category which is can be built by first freely adjoining some objects,
then some $1$-morphisms, then some $2$-morphisms, and so forth, up through $n$-morphisms
and then stop, then there is an explicit geometric des... | 14 | https://mathoverflow.net/users/7721 | 49738 | 31,287 |
https://mathoverflow.net/questions/49733 | 2 | Let $V$ be a finite dimensional vector space over $\mathbb{R}$, and $C\subset V$ a convex cone of the form $C=\mathbb{R}\_{\geq0}v\_i$ for finitely many $v\_i$'s in $V$. How can one describe the stabilizer of $C$ in $GL(V)$?
Here one naturally defines the stabilizer of $C$ to be $GL(C)$ consisting of elements $g\in G... | https://mathoverflow.net/users/9246 | stabilizer of convex cones in a linear space | As Willie already said, you should look at the action of your group on the the extreme rays of C. Look at the kernel of this action, K.
The group GL(C)/K is then finite (and bounds on its order
can be derived from the fact that it will be a permutation action, that is realised in
a subspace of certain dimension...)
... | 3 | https://mathoverflow.net/users/11100 | 49744 | 31,290 |
https://mathoverflow.net/questions/49740 | 17 | Fix a characteristic zero ground field. One can easily check that if $\mathfrak g$ is a simple Lie algebra, then the trilinear map map $\omega$ given by $$\omega(x,y,z)=B([x,y],z),$$ with $B$ the Killing form, represents a generator of $H^3(\mathfrak g,k)$.
Now, $H^\bullet(\mathfrak g,k)$ is an exterior algebra on a... | https://mathoverflow.net/users/1409 | Generators of the cohomology of a Lie algebra | The answer to your question is yes, at least for the classical Lie types. You can find the formulas in Section 6.19 of the book *Connections, curvature, and cohomology, Volume III: Cohomology of principal bundles and homogeneous spaces* by Greub, Halperin, and Vanstone (Pure and Applied Mathematics, Vol. 47-III. Academ... | 7 | https://mathoverflow.net/users/7932 | 49745 | 31,291 |
https://mathoverflow.net/questions/49731 | 30 | I've agreed, perhaps unwisely, to give a talk to Philosophers about string theory.
I'd like to give the philosophers an overview of the status and influence of string theory in physics, which I feel competent to do, but I would also like to say something about the influence it has had in mathematics where I am on les... | https://mathoverflow.net/users/10475 | The influence of string theory on mathematics for philosophers. | Dear Jeff, string theory has had a colossal influence on the renewal of enumerative geometry, a two century old branch of algebraic geometry inextricably linked to intersection theory.
Here is a telling anecdote.
Ellingsrud and Strømme, two renowned specialists in Hilbert Scheme theory, had calculated the number of r... | 30 | https://mathoverflow.net/users/450 | 49747 | 31,293 |
https://mathoverflow.net/questions/49695 | 13 | I read in an article of Erdős ("Extremal problems in number theory") that he had a proof of the multiplicative version of the [Erdős-Turán conjecture](https://mathoverflow.net/questions/43995/are-any-good-strategies-known-for-erdos-turan-conjecture-on-additive-bases-of-ord). The statement of this theorem is
>
> Let... | https://mathoverflow.net/users/2384 | Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture? | Yes.
P. Erdõs: On the multiplicative representation of integers, Israel J. Math. 2 (1964), 251--261 (see: <http://www.renyi.hu/~p_erdos/1964-20.pdf> )
A somewhat different proof is given in:
Nešetřil, Rödl, Two proofs in combinatorial number theory. Proc. Amer. Math. Soc. 93 (1985), no. 1, 185–188.
| 14 | https://mathoverflow.net/users/630 | 49760 | 31,300 |
https://mathoverflow.net/questions/49759 | 23 | If we have a map p: X --> Y of topological spaces, we can make a definition expressing that the topological type of the fibers of p varies continuously (edit: better to say "locally constantly", thanks Dave) with the base: we can say that p is a fiber bundle.
My question is, can we capture this notion algebro-geometr... | https://mathoverflow.net/users/3931 | Definition of fiber bundle in algebraic geometry | ~~OK, let me venture to give a definition. Say that a morphism $f:X\to Y$, of varieties over
a field, is an algebraic fibration if there exists a factorization
$X\to \overline{X}\to Y$, such that that the first map is an open immersion, and the
second map is proper and there exists a partition into Zariski locally clo... | 11 | https://mathoverflow.net/users/4144 | 49766 | 31,304 |
https://mathoverflow.net/questions/49764 | 6 | This question is about the relation between the notions of boundary link and ribbon link.
For the definition of ribbon link see: [ribbon links - counterexamples](https://mathoverflow.net/questions/42293/ribbon-links-counterexamples).
An n-component link $L=L\_1\cup\dots\cup L\_n$ is said to be a boundary link if th... | https://mathoverflow.net/users/5001 | Boundary links and ribbon links. | The answer to your first question is no. There are non-ribbon boundary links whose components are unknotted! Indeed the Bing double of a knot is a boundary link with unknotted components, but it has recently been proven by several authors that the Bing double of the figure-8 knot is not slice (hence not ribbon.) See fo... | 6 | https://mathoverflow.net/users/9417 | 49767 | 31,305 |
https://mathoverflow.net/questions/49786 | 4 | is every prime p equals another prime p' plus or minus a power of 2? p=p'+/-2^n? are there infinitely many primes not of this form?
| https://mathoverflow.net/users/14726 | form of primes:prime plus a power of 2? | 127 and 331 are counterexamples. It was a conjecture of Polignac that every odd number can be written as a sum of an odd prime and a power of two, but many counterexamples have been found. They are called ["obstinate numbers"](http://oeis.org/A133122). Erdos has proved that there is an infinite arithmetic progression o... | 14 | https://mathoverflow.net/users/2384 | 49787 | 31,314 |
https://mathoverflow.net/questions/49688 | 14 | For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with *pointwise* multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform algebra (since it has idempotents of arbitrarily large norm).
It has been known since the 1970s that there exists a... | https://mathoverflow.net/users/763 | "Explicit" embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras | You can manage to get $m(n)=(n+2)2^{n}$ (this can certainly be made a bit better). By your remark, it is enough to construct projections $E\_1,\dots,E\_n$ in $M\_{n+2}(\mathbb C)$ such that:
1. $E\_i E\_j=0$ for $i \ne j$
2. $\|E\_i\|\leq 3$
3. $\| \sum s\_i E\_i\| \geq |\sum s\_i|$ for any complex numbers $s\_1,\dot... | 11 | https://mathoverflow.net/users/10265 | 49788 | 31,315 |
https://mathoverflow.net/questions/49798 | 0 | What should I call a poset with the property that each element has AT MOST ONE
predecessor?
(I'm actually interested in the special case in which there are no infinite descending chains.)
CLARIFICATION: I mean each element has a unique immediate predecessor.
As you go higher up there can be branching, but nev... | https://mathoverflow.net/users/3634 | Terminology for posets. | A partial order with no infinite descending chains is said to be *well-founded*. Every well-founded partial order admits an ordinal ranking function, an assignment of nodes in the order to ordinals, such that higher nodes get larger ordinals. One can therefore speak of the ordinal rank of a node or the height of the wh... | 5 | https://mathoverflow.net/users/1946 | 49801 | 31,321 |
https://mathoverflow.net/questions/49797 | 8 | As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. However, I was not able to find this conjecture in the literature. I would be deeply grateful for any references or det... | https://mathoverflow.net/users/2191 | The conjectural relation between mixed motivic sheaves and the perverse t-structure. | For triangulated category of geometric motives over a regular scheme $S$, the $\ell$-adic realisation has been constructed by Florian Ivorra in his thesis. I think the functor is expected to be t-exact for the motivic and perverse t-structure but don't know if it has been explictly written as a conjecture. There is als... | 1 | https://mathoverflow.net/users/1985 | 49807 | 31,323 |
https://mathoverflow.net/questions/48748 | 10 | Given a smooth complex algebraic variety, the Riemann-Hilbert-correspondence tells us, that the category of perverse sheaves is equivalent to the category of regular, holonomic D-modules.
However not every interesting holonomic D-module is regular. For example the solution sheaves of all the $D\_{\mathbb A^1}$-module... | https://mathoverflow.net/users/2837 | Relation between holonomic D-modules and perverse sheaves | The answer is yes but it's not easy. You need additional data to describe the irregular part of your connexion. These are known as Stokes structures. Very loosely it's a filtration of your sheaf of solutions according to their growth in a given sector. Very recently, Claude Sabbah has written lecture notes on the subje... | 6 | https://mathoverflow.net/users/1985 | 49809 | 31,325 |
https://mathoverflow.net/questions/49799 | 21 | Given a graph $G(V,E)$ whose edges are colored in two colors: red and blue.
Suppose the following two conditions hold:
* for any $S\subseteq V$, there are at most $O(|S|)$ red edges in $G[S]$
* for any $S\subseteq V$, if $G[S]$ contains no red edges, then it contains $O(|S|)$ blue edges
My question is: can we conc... | https://mathoverflow.net/users/5572 | A graph with few edges everywhere | I think one can push through the probabilistic arguments of Tim Gowers and Fedor Petrov in the general case, as follows.
Let $c$ be a constant such that the number of red edges in $G[S]$
is at most $c|S|$ for every $S \subseteq V(G)$. One can order the vertices of $G$: $v\_1, v\_2, \ldots, v\_n$, so that every verte... | 12 | https://mathoverflow.net/users/8733 | 49830 | 31,339 |
https://mathoverflow.net/questions/49835 | 12 | Let $X$ be a proper smooth algebraic space over $\mathbb C$ (which amounts, due to Artin, to giving a Moishezon space: a compact complex manifold whose dimension equals the transcendence degree of its field of meromorphic functions). I'd like to know if the classical Hodge theory holds on the cohomology of $X$ (e.g. de... | https://mathoverflow.net/users/370 | Hodge structures on algebraic spaces | If the Hodge to de Rham sequence degenerates for a smooth compact complex manifold $M$, it degenerates for any smooth compact manifold bimeromorphic to $M$, see theorem 5.22 of Deligne, Griffiths, Morgan, Sullivan, Real homotopy theory of K\"ahler manifolds (theorem 5.22 is about the $dd^c$ lemma, but this is equivalen... | 7 | https://mathoverflow.net/users/2349 | 49839 | 31,342 |
https://mathoverflow.net/questions/49848 | 4 | It seems to me that these 3 algebraic systems are closely related, but it always seems to be Noetherian rings rather than Noetherian domains appearing, and conversely I rarely seem to see principal ideal rings or Dedekind rings.
| https://mathoverflow.net/users/4692 | Why are principal ideal domains and Dedekind domains prominent, but I always seem to see Noetherian rings rather than Noetherian domains? | Dear Negative refraction,
I would guess that it reflects the particular literature you are looking at. If you were to look at algebraic geometry literature, you would very often see the following line (or a variant thereof)): let $U =$ Spec $A$ be open affine in the irreducible variety $X$. The ring $A$ will then be... | 8 | https://mathoverflow.net/users/2874 | 49851 | 31,345 |
https://mathoverflow.net/questions/48223 | 7 | Motivation
----------
Suppose that we need to estimate the median from a normal distribution with known variance. One non-parametric approach is to use the sample median as an estimator. However, this does not take advantage of our distributional assumptions. Exploiting, the symmetry of the normal distribution, we ca... | https://mathoverflow.net/users/10203 | Parametric vs Non-parametric Estimation of Quantiles | Just in case someone is following, I want to post a somewhat negative answer to my second question. I found an example that satisfies the assumptions, and achieves an efficiency arbitrarily close to 1.
The example is inspired on Laplace distribution with an unknown location parameter $\theta$, and p.d.f. $f(x|\theta... | 2 | https://mathoverflow.net/users/10203 | 49856 | 31,348 |
https://mathoverflow.net/questions/49794 | 10 | Let $K$ be a finite extension of $\mathbb{Q}\_p$ and $L$ be a finite Galois extension of $K$. Let $V$ be a $p$-adic representation of $G\_L$ Let $D$ be the $(\varphi, \Gamma)$-module associated to $V$ by Fontaine's functor. Is there an easy way to describe the $(\varphi, \Gamma)$-module associated to $\operatorname{Ind... | https://mathoverflow.net/users/10001 | Induced representations and $(\varphi, \Gamma)$-modules | If $L/K$ is unramified, then the answer is exactly what Matt said. In the general case, you have to take into account the fact that $\Gamma\_K$ is larger than $\Gamma\_L$ and the construction is given in 2.2 of Ruochuan Liu's "Cohomology and Duality for (phi,Gamma)-modules over the Robba ring", see <http://arxiv.org/ab... | 11 | https://mathoverflow.net/users/5743 | 49858 | 31,350 |
https://mathoverflow.net/questions/49850 | 3 | Just as we know,
$w\_\infty$:=span {${z^\alpha }\partial \_z^\beta|\alpha,\beta\in\mathbb{Z}, \beta\geq0$ }.
But, what's the name of the following algebra,
span {$\{{z^{\alpha\_1}}{y^{\alpha\_2}}\partial \_z^{\beta\_1}\partial \_y^{\beta\_1}|\alpha\_i,\beta\_i\in\mathbb{Z}, \beta\geq0\}$ }?
Is it isomorphic to $w\_... | https://mathoverflow.net/users/5705 | What's the name of an algebra? is it isomorphic to $w_\infty \times w_\infty$? | If I'm not mistaken, this is the associative algebra of algebraic differential operators on the torus $\mathbb{G}\_{m,\mathbb{C}}^2$. It is the tensor product of two copies of $w\_\infty$, not the direct product. That is, you should replace $\times$ with $\otimes$.
| 3 | https://mathoverflow.net/users/121 | 49860 | 31,351 |
https://mathoverflow.net/questions/49859 | 4 | Hello,
Consider some logical statement (I am talking about natural numbers all the way). P(x, y, z) is a computable statement:
For all x: There exists an y: For all z: P(x,y,z) is true.
I suppose this would have a Kleene level of 3.
Now, you could not consider *all* x, y and z, but only the values from zero to ... | https://mathoverflow.net/users/11506 | Why does the Kleene Hierarchy not collapse? | You're right that the statement $\varphi(a,q,v,w)$ defined by $\forall x<a+q \,\, \exists y<a+v \,\, \forall z<a+w [P(x,y,z)]$ can be checked by a Turing machine. If I read you correctly, you're wondering whether (1) $\forall x \exists y \forall z P(x,y,z)$ is generally equivalent to (2) $\forall a \exists q,v,w \varph... | 6 | https://mathoverflow.net/users/4137 | 49861 | 31,352 |
https://mathoverflow.net/questions/49771 | 4 | I am interested in calculating properties of a continuous-time random walk problem which I believe is a type of semi-Markov process.
I have states of the form $n\_\pm \in \mathbb{Z} \times \{ +, -\}$. For a state $n\_\pm$, I have time-dependent transition probabilities $p\_{\pm+}(t)$ and $p\_{\pm-}(t)$ for jumping to... | https://mathoverflow.net/users/11657 | Properties of a continuous-time semi-Markov process as t -> \infty | I hesitated a bit whether to use another answer window or to edit the old one but finally decided in favor of a new window. If moderators think it is a bad idea, they are welcome to merge.
Also, it is 6:30AM and it promises to be quite a busy day, so I'll tell you what and how to count but will not attempt to do the... | 6 | https://mathoverflow.net/users/1131 | 49876 | 31,359 |
https://mathoverflow.net/questions/49866 | 33 | I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For [Applications of periodic continued fractions](https://mathoverflow.net/questions/49930/applications-of-periodic-continued-fractions) I have made a special topic.)
1) (Trivial) Analysis of Euclidean algorithm (a... | https://mathoverflow.net/users/5712 | Applications of finite continued fractions | In knot theory continued fractions are used to classify rational tangles. Conway proved that
two rational tangles are isotopic if and only if they have the same fraction. This is
proved by Kauffman in <http://arxiv.org/pdf/math/0311499.pdf>. The paper also contains all the basic definitions and I think it can be read ... | 22 | https://mathoverflow.net/users/5001 | 49877 | 31,360 |
https://mathoverflow.net/questions/21757 | 1 | Let $X = {x\_1, ..., x\_N}$ be a finite subset of $R^n$ and let $p$ and $q$ be any polynomials of degree $k$ or less. X is called $\underline{P\_k-unisolvent}$ if $p(x\_j) = q(x\_j)$ ($j = 1, ..., N$) implies that $p=q$; i.e. evaluation of a polynomial on $X$ uniquely determines that polynomial.
In one dimension, uni... | https://mathoverflow.net/users/3776 | References regarding unisolvent sets | The term "unisolvent" is inspired by the much more classical definition involving functions. (e.g. Philip Davis - Interpolation and Approximation, and results by B. Polster)
A unisolvent family of functions contains a unique solution to the interpolation problem given a collection of points;
Whereas a unisolvent point ... | 0 | https://mathoverflow.net/users/3776 | 49884 | 31,363 |
https://mathoverflow.net/questions/49885 | 17 | For an affine algebraic group $G$ it's often convenient (and harmless) to work concretely over an algebraically
closed field of definition $k$ while identifying $G$ with its group of rational points over $k$.
Once in a while, however, results obtained over $k$ need to be compared with results over
a bigger algebraicall... | https://mathoverflow.net/users/4231 | Comparing algebraic group orbits over big and small algebraically closed fields | I think this will work. There are a finite number of orbits of the action of $G$
on $X$ precisely when there is an open orbit and a finite number of orbits on
the complement of the orbit. Hence, it is enough to show that if there is an
open orbit of a point over the smaller field precisely when there is an open
orbit o... | 10 | https://mathoverflow.net/users/4008 | 49890 | 31,365 |
https://mathoverflow.net/questions/49775 | 9 | In classical logic plus ZF, the field of real numbers admits infinitely many isomorphic realizations as a numeral system --- as the radix varies. The intuitionistic status of these systems seems less clear however. First of all, the notion of "field" has several
distinct intuitionistic interpretations (e.g. non-zero im... | https://mathoverflow.net/users/10909 | Radix notation and toposes | No, and actually, you cannot realise any non-trivial equivalence relation this way. If any of the (non-trivial) pairs of radix systems are isomorphic, they all are. It is "well-known" that not every real number has a decimal expansion. This extends to this situation.
Specifically, for any pair n and m (unless one div... | 6 | https://mathoverflow.net/users/6787 | 49904 | 31,375 |
https://mathoverflow.net/questions/49906 | 1 | I'm interested in $\theta(N):=\int\_0^1 (1-x)^{N-1} e^{xN} dx$. I'd like to show that $\theta(N)\sim c/\sqrt{N}$ as $N\to\infty$ and determine $c$. Any ideas?
| https://mathoverflow.net/users/11697 | limit of definite integral as $N \to \infty$ | Denote by $I$ your integral. Then,
$I = e^N \int\_0^1 {x^{N - 1} e^{ - xN} \,{\rm d}x} = \frac{{\Gamma (N)e^N }}{{N^N }}\int\_0^N {\frac{{x^{N - 1} e^{ - x} }}{{\Gamma (N)}}\,{\rm d}x}.$
Now, if $X\_1,\ldots,X\_N$ are independent and identically distributed exponential(1) random variables, then their sum $X\_1 + \cdots... | 9 | https://mathoverflow.net/users/10227 | 49909 | 31,376 |
https://mathoverflow.net/questions/49913 | 12 | I am asking [my question](https://math.stackexchange.com/questions/8332/roots-of-unity-and-field-extensions) here, since it's been voted up a fair bit on math.SE, but without answers, so it may be harder than I assumed it was.
Can we always break an arbitrary field extension $L/K$ into an extension $F/K$ in which the... | https://mathoverflow.net/users/1916 | Factoring a field extension into one which adds no roots of unity, followed by one which adds only roots of unity | The answer is no. The idea is to consider a tamely ramified extension of a local field. The remaining paragraphs provide details of a proof.
Take K=ℚp. Let M be a degree d unramified extension of K. Let L be an extension of M obtained by adjoining a n-th root of $pa$ for some $a$ of norm 1 in M that is not in ℚp. We ... | 10 | https://mathoverflow.net/users/425 | 49914 | 31,379 |
https://mathoverflow.net/questions/49923 | 3 | I'm re-reading a paper of Stevo Todorcevic's entitled "Localized Reflection and Fragments of PFA" and there's a claim in the proof of one of the lemmas that I thought I understood but now I'm not so sure. The claim is this:
>
> Suppose $0^{\sharp}$ does not exist, and let $a \in L\_{\omega\_2}$, $\varphi$ a formula... | https://mathoverflow.net/users/7521 | A question about 0# and truth in levels of the L hierarchy | Amit, I do not think this is exactly what Stevo is claiming. He writes:
>
> Let $\varphi$ be a given formula of set theory and suppose that for some
> $a\in L\_{\omega\_2}$ there is $\theta$, a regular cardinal in $L$ such that $L\_\theta\models\varphi(a)$. We need to find $\theta'\lt\omega\_2$, a regular cardina... | 5 | https://mathoverflow.net/users/6085 | 49926 | 31,383 |
https://mathoverflow.net/questions/49916 | 11 | My (perhaps inaccurate) impression is there are many competing definitions of the notion of a (weak) $n$-category, none of which are generally accepted. I've run across several properties such definitions should satisfy, e.g. that weak $n$-groupoids should model homotopy $n$-types, or that the collection of $n$-categor... | https://mathoverflow.net/users/6950 | What properties should a good definition of (weak) $n$-category satisfy? | I've just heard a talk by Julia Bergner on the subject, reporting on her ongoing work with Charles Rezk.
Here's the list of properties that they are trying to establish for their model of $(\infty,n)$-categories.
* For each $n>0$, the category of $(\infty,n)$-categories is a cartesian model category (i.e. it's a mod... | 7 | https://mathoverflow.net/users/5690 | 49936 | 31,388 |
https://mathoverflow.net/questions/49811 | 3 | Hi, I am interested in the set $\mathbb A-\mathbb A^\times$ i.e. the complement of ideles in the adele ring of a number field.
Is it measurable, and what is its volume, with respect to the standard measure of adeles?
("standard" means the same as in Tate's thesis)
Thank you.
| https://mathoverflow.net/users/4245 | Measure of "adeles minus ideles" | This is a bookkeeping post, since the answer seems to have been resolved in the comments. Somebody please vote this up once so this question leaves the "unanswered" queue.
Shenghao's answer is essentially that you can view the ideles as a countable union of translates of $\widehat{\mathbb{Z}}^\times$, which has measu... | 10 | https://mathoverflow.net/users/121 | 49941 | 31,390 |
https://mathoverflow.net/questions/49942 | 2 | Given a complex of vector spaces $M$ , is it possible to find another complex $\tilde{M}$ such that $H^{i}(\tilde{M})=0$ for $i > 0$ and with a (term-wise) surjection $\tilde{M} \rightarrow M$
such that $H^{i}(\tilde{M}) \rightarrow H^{i}(M)$ is surjective for all $i \leq 0$. Even better if these maps on cohomology are... | https://mathoverflow.net/users/11715 | Killing cohomology of a complex | Let $\tau^{\geq 1}M$ be the complex $\cdots \to 0 \to M^{0} / \ker{d^{0}} \to M^{1} \to M^{2} \to \ldots$. The obvious map $f: M \to \tau^{\geq 1}M$ yields a triangle
\[
M \to \tau^{\geq 1}M \to C(f) \to M[1]
\]
and shifting this back by $[-1]$ we get a map $C(f)[-1] \to M$ with the desired properties.
---
Superf... | 3 | https://mathoverflow.net/users/11081 | 49950 | 31,394 |
https://mathoverflow.net/questions/49946 | 1 | Last week I considered again *principal curvature (pc)* and *principal curvature directions (pcd)* of a, for the sake of simplicity, 2-manifold embedded in 3-space. In this simple case, the pc and pcd of at a point are the eigenvalues and eigenvectors of the shape operator. The magnitude of the pc's corresponds to the ... | https://mathoverflow.net/users/8047 | Principal curvatures and curvature directions | Unless there's an additional constraint on the defintion of the principal curvature directions, being just defined as eigenvectors means their magnitude is arbitrary and so meaningless.
| 5 | https://mathoverflow.net/users/11640 | 49951 | 31,395 |
https://mathoverflow.net/questions/49883 | 10 | It's a question I've been thinking about but I can't find an easy answer. I think it will be interesting. Can there be a countable collection of real valued functions $f\_1, f\_2 , ... $ such that for any subset $K$ of $\mathbb R$ of cardinality continuum, the set of those $n$ such that $f\_n(K)$ is not the whole of $\... | https://mathoverflow.net/users/4903 | Set theoretic question about real valued functions | I have three observations.
First, I have noticed that there can be no such sequence of
functions $f\_n$, if one insists that every $f\_n$ is a
measurable function. In particular, there can be no sequence of Borel functions with your property. To see this, suppose that
$f\_n:\mathbb{R}\to\mathbb{R}$ is a countable se... | 6 | https://mathoverflow.net/users/1946 | 49962 | 31,400 |
https://mathoverflow.net/questions/49963 | 3 | Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The group of diffeomorphisms preserving the marked points acts on this set. What are the orbits? How many are there? Equivalent... | https://mathoverflow.net/users/184 | Is the Action of the mapping class group transitive on embedded arcs? | There is only one orbit.
Suppose you have two such arcs $\lambda, \lambda'$. Let $S\_\lambda$ be obtained from $S$ by removing a regular neighborhood of $\lambda$: the regular neighborhood is a disc, containing the two marked points in its interior. The two surfaces $S\_\lambda$ and $S\_{\lambda'}$ so obtained have ... | 7 | https://mathoverflow.net/users/6205 | 49964 | 31,401 |
https://mathoverflow.net/questions/49960 | 22 | The absolute Galois group $G\_{\mathbb Q}=\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$, as a profinite group, encodes a lot of things: the whole lattice of number fields (closed subgroups of finite index), Galois extensions (normal subgroups), abelian extensions etc. Is it possible to recognize *unramified* abelian extension... | https://mathoverflow.net/users/3132 | Are class numbers encoded in the absolute Galois group of ${\mathbb Q}$? | Dear Tim,
As you're probably aware, this is part of the 'anabelian' etcetera.
It suffices to recover all intertia subgroups $I\_v\subset H$, because their union will then be a normal subgroup $N$ such that $H/N$ is the Galois group of the maximal extension of $K$ unramified everywhere. We can get the ideal class g... | 17 | https://mathoverflow.net/users/1826 | 49967 | 31,403 |
https://mathoverflow.net/questions/49975 | 2 | Suppose that a game is played on an $n \times n$ board as follows. There are two players, Player 1 has (only) $Q$ queens and Player 2 has only $K$ knights. Suppose that $Q, K \leq n/3$. The game is played as per usual in chess, with Player 1 going first. The objective is to capture all of the other players' pieces. Now... | https://mathoverflow.net/users/10898 | A random variable in a game of knights and queens | It is standard to show that $X$ has exponential decay: for any position, there is a fixed positive probability that the game will terminate within the next (say) 10n steps. In particular, all moments of $X$ are finite.
| 6 | https://mathoverflow.net/users/1061 | 49980 | 31,408 |
https://mathoverflow.net/questions/49977 | 1 | Given that simply stipulating positive upper density is not sufficient to guarantee that all but finitely many members are in an arithmetic progression of length 3, that there indeed exists sets of integers with positive density that contain infinitely many elements that are not contained in an arithmetic progression o... | https://mathoverflow.net/users/10898 | Do there exist sets of integers with arbitrarily large upper density which contains infinitely many elements that are not in an arithmetic progression of length 3? | No.
If $a\in A$ is not in an AP of length $3$ then $A$ contains at most $n/2$ terms from $(a,a+n]$. So $A$ has density at most $1/2$. Density $1/3$ is easy to construct: take numbers of the form $3n$ or $4^n$.
| 1 | https://mathoverflow.net/users/9422 | 49987 | 31,411 |
https://mathoverflow.net/questions/49970 | 7 | Let $\mathcal{A}\_{g,D}$ be the moduli space of abelian varieties of dimension $g$ and polarization $D$ of type $(d\_1, \ldots, d\_g)$.
Let $\mathcal{M}$ be the moduli space parametrizing pairs $(A, \mathcal{L})$, where $A \in \mathcal{A}\_{g,D}$ and $\mathcal{L}$ is a non-trivial $2$-torsion line bundle on $A$, i.e.... | https://mathoverflow.net/users/7460 | $2$-torsion line bundles on abelian varieties | Let's stick to $D=(1,2)$. Associated to any point of $\mathcal{M}$ (say, a complex point) we have an abelian surface $A$ and an isogeny $\phi:A\to A'$ where $A'=\mathrm{Pic}^0(A)$ is the dual. The kernel of $\phi$, in our case, is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. So, the image of $A[2]$ in $A'$ is a canonica... | 7 | https://mathoverflow.net/users/7666 | 49988 | 31,412 |
https://mathoverflow.net/questions/49995 | 8 | Let $\pi:E\to X$ be a holomorphic vector bundle of degree 0 over a compact Riemann surface $X$. Why does $E$ admit a flat connection. I could work this out in the case of line bundles, where one starts with the natural logarithmic connection on $\mathscr{O}(\sum\_{i=0}^{k}n\_iP\_i)$ (here $\sum\_{i=0}^{n}n\_i=0$) and m... | https://mathoverflow.net/users/11395 | Flat connections on Bundles of degree 0 on a compact Riemann surface | Rex: A holomorphic bundle doesn't always admit a flat connection. You need to assume further that each of its indecomposable pieces has degree 0. This is the result of Weil [J. Math. Pures Appl. (9) 17 (1938), 47--87] and Atiyah [Trans. Amer. Math. Soc. 85 (1957), 181–207].
| 8 | https://mathoverflow.net/users/9867 | 50000 | 31,418 |
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