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https://mathoverflow.net/questions/62596 | 11 | Which finite subsets $S \subset \mathbb{N}$ have the following property : every countable group $G$ embeds into a finitely generated group $\Gamma$ such that $H\_i(\Gamma;\mathbb{Z})=0$ for all $i \in S$.
The only positive answer I know here is that $S=\{1\}$ works since every countable group can be embedded into a s... | https://mathoverflow.net/users/317 | Embedding groups into groups with some vanishing homology groups | See Baumslag, G.; Dyer, E.; Miller, C. F.
On the integral homology of finitely presented groups.
Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 321–324, and the full version Baumslag, G.; Dyer, E.; Miller, C. F., III On the integral homology of finitely presented groups. Topology 22 (1983), no. 1, 27–46. Lemma 4 in pa... | 5 | https://mathoverflow.net/users/nan | 62814 | 38,840 |
https://mathoverflow.net/questions/62812 | 3 | Where can I find Grothendieck's "Pursuing Stacks"/"A la poursuite des champs" and "Esquisse d'un programme"?
| https://mathoverflow.net/users/7626 | References for Grothendieck's "Pursuing Stacks" and "Esquisse" | Really Martin's [comment](https://mathoverflow.net/questions/62812/references-for-grothendiecks-pursuing-stacks-and-esquisse#comment157643_62812) should be the answer, but note Wikipedia also gives:
Alexander Grothendieck, 1984. "Esquisse d'un Programme", (1984 manuscript), finally published in Schneps and Lochak (19... | 7 | https://mathoverflow.net/users/3502 | 62823 | 38,843 |
https://mathoverflow.net/questions/62819 | 5 | Could anyone recommend a reference to a canonical form for a pair of quadratic forms over R (not necessarily positively definite)? This is probably related to the Weierstrass elementary divisors (but it is not so easy to find information about them on the web). To be more precise, the case when A=diag(1,1,1,1,-1) and B... | https://mathoverflow.net/users/14639 | Canonical form for a pair of quadratic forms | In a standard exposition (I.M. Gelfand, Lectures on Linear Algebra, or A. I. Maltsev, Foundations of Linear Algebra) it is required that one of the forms is positive definite, and this cannot be dropped. Some general results in terms of elementary divisors are given in Maltsev's book too. A little Google search gives a... | 3 | https://mathoverflow.net/users/12205 | 62827 | 38,846 |
https://mathoverflow.net/questions/62820 | 24 | What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
* n=3. The formula for solutions of $x^2+y^2=z^2$ [4]:
$x=d(p^2−q^2)$,
$y=2dpq$,
$z=d(p^2+q^2),$
where p,q,d are arbitrary polyn... | https://mathoverflow.net/users/14639 | Pythagorean 5-tuples | One way to generate integer solutions is as follows: Let $ v = (p + qi + rj + sk)$, where $p,q,r,s$
are rational integers and $\{1,i,j,k\}$ is the usual $\mathbb{R}$-basis for the algebra of quaternions. Let $v^{\prime}= (p -qi -rj - sk)$ and
$w = vv^{\prime} = (p^{2}+q^{2}+r^{2}+s^{2}).$
If we expand $v^{2}$ in the ... | 11 | https://mathoverflow.net/users/14450 | 62829 | 38,848 |
https://mathoverflow.net/questions/57232 | 5 | Given a Heegaard splitting of genus $n$, and two distinct orientation preserving homeomorphisms, elements of the mapping class group of the genus $n$ torus, is there a method which shows whether or not these homeomorphisms, when used to identify the boundaries of the pair of handlebodies, will produce the same $3$-mani... | https://mathoverflow.net/users/nan | Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus | The original question was "Is there an algorithm that, given two genus g Heegaard splittings, decides if the resulting two manifolds are homeomorphic?" The answer to this question is "yes" but doesn't have anything to do with Heegaard splittings.
In theory the solution to the Geometrization Conjecture gives an algor... | 5 | https://mathoverflow.net/users/1650 | 62830 | 38,849 |
https://mathoverflow.net/questions/62816 | 8 | I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.
The [explicit formula of Guinand and Weil](http://en.wikipedia.org/wiki/Explicit_formula#Weil.27s_explicit_formula) can be written in the following way:
*For 'nice' g* (i.e. in $C\_c^\infty(\mathbb{R... | https://mathoverflow.net/users/5621 | The Guinand-Weil explicit formula without entire function theory | The Riemann zeta function is given for $Re s>1$
$$\zeta(s) = \prod\limits\_{p \; prime} ( 1-p^{-s}) $$
This product converges absolutely in $Re s >1$, hence it does not vanish in $Re s>1$. Actually the product also converges locally uniformly, which implies that $\zeta(s)$ is holomorphic for $Re s>1$.
The functio... | 10 | https://mathoverflow.net/users/10400 | 62836 | 38,852 |
https://mathoverflow.net/questions/62818 | 40 | It is well known that intuitive set theory (or naive set theory) is characterized by having paradoxes, e.g. Russell's paradox, Cantor's paradox, etc. To avoid these and any other discovered or undiscovered potential paradoxes, the ZFC axioms impose constraints on the existense of a set. But ZFC set theory is build on m... | https://mathoverflow.net/users/5072 | The sets in mathematical logic | I have been asked this question several times in my logic or set theory classes.
The conclusion that I have arrived at is that you need to assume that we know how to deal
with finite strings over a finite alphabet. This is enough to code the countably many
variables we usually use in first order logic (and finitely or... | 39 | https://mathoverflow.net/users/7743 | 62838 | 38,854 |
https://mathoverflow.net/questions/62783 | 3 | The [Orlicz space](http://en.wikipedia.org/wiki/Birnbaum%25E2%2580%2593Orlicz_space) associated to the convex function $e^{x^2}-1$ arises frequently in probabilistic problems (being in this space implies that a function has sub-Gaussian tails). Does this space have a more concise name?
| https://mathoverflow.net/users/630 | Does the Orlicz space associated to $e^{x^2}-1$ have a name? | One often denotes $\psi\_\alpha (x) = e^{x^\alpha} - 1$ for $\alpha \ge 1$, and the corresponding Orlicz space is $L\_{\psi\_\alpha}$. An estimate on the $L\_{\psi\_\alpha}$ norm is also often referred to simply as a $\psi\_\alpha$ estimate. Of course, the case $\alpha = 2$ which you asked about is the one most often o... | 3 | https://mathoverflow.net/users/1044 | 62840 | 38,855 |
https://mathoverflow.net/questions/62758 | 6 | Just a silly little question which arose in connection with infinite Galois groups and their Krull topology:- can a given abstract group be endowed with distinct, non-homeomorphic, profinite topologies ? (I asked this question several years ago on the Topology Q+A and was told the question is undecidable and has someth... | https://mathoverflow.net/users/13462 | Distinct, non-homeomorphic, profinite topologies on a given abstract group ? | Yes.
I have classified some abelian examples: there are uncountably many pairwise non-homeomorphic pro-$p$ topologies that can be placed on the (unrestricted) product of any countable collection of cyclic $p$-groups of unbounded exponent.
The results are presented here, but I am in the process of redrafting <http... | 4 | https://mathoverflow.net/users/4100 | 62841 | 38,856 |
https://mathoverflow.net/questions/27346 | 16 | One of the consequences of Seymour's characterization of regular matroids is the existence of a polynomial time recognition algorithm for totally unimodular matrices (i.e. matrices for which every square sub-determinant is in {0, 1, -1}).
But has anyone actually implemented it?
| https://mathoverflow.net/users/1492 | Has anyone implemented a recognition algorithm for totally unimodular matrices? | **EDIT.** Walter and Trümper have [announced on arXiv](https://arxiv.org/abs/1202.4061) their implementation, with source code available, of two methods for testing total unimodularity. Their paper describes the technical details of the implementation / algorithm, and also provides several experimental results.
---... | 10 | https://mathoverflow.net/users/8430 | 62842 | 38,857 |
https://mathoverflow.net/questions/62815 | 6 | Let $s$ be an integer greater than 1. For each natural number $n$, let $\omega(n)$ be the number of prime divisors (counted with multiplicities) of $n$ modulo $s$. For $i \in \{0,1,\dots, s-1\}$ and a positive integer $k$, set
$$c\_i(k) = |\{x \in \{1, 2, \dots, k\} : \omega(x) = i\}|.$$
Is it true that $\frac{c\_i(k)... | https://mathoverflow.net/users/14233 | proportion of positive integers which number of prime divisors has a special remainder. | Yes, this is true.
More precisely, it is known that, for fixed $s$ and $k \to \infty$, one has
`$$ \frac{c_i(k)}{k} = \frac{1}{s} + O(\frac{1}{ (\log k)^A}) $$`
for some $A>0$ .
See 'On the residue class distribution of the number of prime divisors of an integer' by Coons and Dahmen; preprint at <http://arxiv.org/... | 4 | https://mathoverflow.net/users/nan | 62845 | 38,859 |
https://mathoverflow.net/questions/62855 | 8 | Consider the manifold $\mathbb{R^2}\setminus \{0\}$, on which the group of rotation acts. The orbits of the group are the circles centered in the origin, and form a foliation of $\mathbb{R^2}\setminus \{0\}$. This foliation will be denoted by $F\_1$. The foliation $F\_1$ defines univocally another foliation $F\_2$, wit... | https://mathoverflow.net/users/9584 | Orthogonal foliations | There's a distinction to be made between two notions: *foliations* and *distributions*.
A **distribution** is the data, at each point *m* of *M*, of a subspace of *Tm*(*M*). These subspaces are all of the same dimension (say *r*), and depend smoothly on the point *m*, which means that they are generated by *r* smooth... | 15 | https://mathoverflow.net/users/5690 | 62867 | 38,870 |
https://mathoverflow.net/questions/62868 | 4 | Inspired by [this question](https://mathoverflow.net/questions/23547/), I would like to know what is the longest known sequence of consecutive zeros in Pi (in base 10).
So far the longest I have found is the sequence of 8 zero's occurring in position 172,330,850 after the decimal point.
If we expand the question t... | https://mathoverflow.net/users/1320 | What is the longest known sequence of consecutive zeros in Pi? | There is a sequence of 12 zeroes starting at position 1755524129973; There is a sequence of 13 eights starting at position 2164164669332. You can see more statistics in [Fabrice Belard's web pages](http://bellard.org/pi/pi2700e9/pidigits.html).
| 8 | https://mathoverflow.net/users/1168 | 62871 | 38,871 |
https://mathoverflow.net/questions/62866 | 42 | What are the recent and new applications of Mathematics in other Sciences ?
Let me try to be more precise about the question:
* By "recent" I mean the last 15 years.
* By "new" I want to exclude the standard answers like cryptography or finance
* By "applications" I mean a mathematical concept (or even a trick) su... | https://mathoverflow.net/users/7031 | Recent Applications of Mathematics | Statistics applied to microarray data in biology. And Bernd Sturmfels and his students have been applying algebraic geometry to this. He wrote a book titled *Algebraic statistics for computational biology*. Biology is a field that will explode in coming decades. Advances in that field will probably capture the public i... | 4 | https://mathoverflow.net/users/6316 | 62876 | 38,875 |
https://mathoverflow.net/questions/62843 | 17 | Let $X$ be a variety. Then, is $X$ path connected? And by path connected, I mean any two closed points $P, Q$ on the variety can be connected by the image of a finite number of non-singular curves.
| https://mathoverflow.net/users/9035 | Path connectedness of varieties | Given any two points on a projective variety, blow them up and re embed the blownup variety in P^N. Then by Bertini, any general linear section of the right codimension will meet the variety in an irreducible curve which also meets both exceptional divisors. Then blowing back down gives an irreducible curve connecting ... | 45 | https://mathoverflow.net/users/9449 | 62883 | 38,881 |
https://mathoverflow.net/questions/52830 | 7 | Hi all,
I am going to give a talk in a seminar about the general theme 'sum of squares'. My interests lie in Differential Geometry, so I recalled that the curvature is a local invariant, an obstruction to the metric being locally a sum of squares of coordinate differentials.
Can some people suggest me some good bo... | https://mathoverflow.net/users/7780 | Curvature and Riemannian metric | This is maybe late for your seminar, but a classical textbook about Riemannian geometry, including relations between curvature and metric tensor, with any signature, is
Riemannian geometry by L.P.Eisenhart
Another very interesting book for you could be
Spaces of constant curvature by J. A. Wolf
Indeed, it seem... | 2 | https://mathoverflow.net/users/8208 | 62894 | 38,886 |
https://mathoverflow.net/questions/62898 | 6 | Suppose we have two differentiable paths $\alpha$ and $\beta$ thru the identity of a Lie group $G$, $\alpha(0)=\beta(0)=e$ the identity element. Denote $\alpha\beta$ the path given by $\alpha\beta(t)=\alpha(t).\beta(t)$ where the dot denotes the group operation. We also have $\alpha\beta(0)=e$.
The paths are differen... | https://mathoverflow.net/users/14540 | Lie group operation and tangent vectors | Here's another way to look at the problem. The derivative of a differentiable map at any point is a linear map of tangent spaces. We have five differentiable maps in play:
1. The "pair of paths" map $(\alpha, \beta): (-\epsilon, \epsilon) \times (-\epsilon, \epsilon) \to G \times G$.
2. The multiplication map $m: G \... | 11 | https://mathoverflow.net/users/121 | 62915 | 38,901 |
https://mathoverflow.net/questions/62918 | 2 | Let $X$ be a [polish space](http://en.wikipedia.org/wiki/Polish_space) (separable completely metrizable topological space). Let $m$ be a probability measure on $X$ and $f:X \rightarrow \mathbb{R}$ a measureable function. I want to show that $f$ is equal a.e. to a second Baire class function (limits of limits of bounded... | https://mathoverflow.net/users/14623 | Measurable function is Baire class 2 almost everywhere | It seems to me that the general case follows from the bounded case. If $f$ is not necessarily bounded, consider $\arctan\circ f$ and represent it as a limit of limits of continuous functions. Then apply $\tan$ to all those functions and get the required representation of $f$. (Since all the limits you care about are po... | 1 | https://mathoverflow.net/users/6794 | 62920 | 38,903 |
https://mathoverflow.net/questions/62916 | 3 | Hi everyone!
I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not be hard, but I am lost...
>
> Question: Suppose $T$ is an invertible
> measure-preserving map of standard probabil... | https://mathoverflow.net/users/14660 | trivial map on $\sigma-$algebra $\mod{}0$ is trivial | If $X$ is a standard probability space then we may assume it to be the disjoint union of an interval with Lebesgue measure and a countable set of atoms. If $p$ is an atom, then by assumption, $T(p) = p$, since $p$ has positive measure. So none of the atoms can be "bad" points. So we may assume that there are no atoms, ... | 4 | https://mathoverflow.net/users/7392 | 62927 | 38,907 |
https://mathoverflow.net/questions/62931 | 2 | Given a closed bounded set $X \subset \mathbb{R}^3$ and two curves $\gamma\_1$ and $\gamma\_2$ in the group of orientation preserving isometries of $\mathbb{R}^3$. Define the sets $X\_1$ and $X\_2$ as the infinite intersections $X\_1 = \bigcap\_{p \in \gamma\_1} pX$ and $X\_2 = \bigcap\_{q \in \gamma\_2} qX$ where $kX$... | https://mathoverflow.net/users/14667 | Measure of infinite intersection of sets | Here's a counterexample in $\mathbb{R}^2$; to get one in $\mathbb{R}^3$ just take the product with $[0,1]$. Let $X=[0,2]\times[0,1]$. Let $f:[0,1]\to[0,1]$ be a continuous function with $f(0)=0$, $f(1)=0$, $f(x)>0$ for $0<x<1$, and $(2-x)(1-f(x))$ strictly decreasing. (If my arithmetic is good, $f(x)=x-x^2$ works, but ... | 1 | https://mathoverflow.net/users/6794 | 62940 | 38,914 |
https://mathoverflow.net/questions/62958 | 4 | I notice there is a strong connection between shellability of simplicial complexes and bistellar flips on these complexes; in particular, adding in a new facet of a shelling induces a bistellar flip on the boundary.
Is it always the case that bistellar flips preserve shellability of a complex? In other words, if I ap... | https://mathoverflow.net/users/14675 | Do bistellar flips preserve shellability? | Obviously not, because there exist non-shellable combinatorial spheres, but any combinatorial $n$-sphere is bistellar-equivalent to the boundary of the $(n+1)$-simplex.
The observation you mentioned is also in the very end of [Lickorish's paper](http://www.math.uic.edu/~kauffman/Lickorish.pdf)
| 6 | https://mathoverflow.net/users/10819 | 62960 | 38,922 |
https://mathoverflow.net/questions/62782 | 11 | Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition dependent on it's location in the grid. This comes from a random conductance model. The theorem that concerns me is a general ... | https://mathoverflow.net/users/934 | Random walk origin return monotinicity | I am not exactly sure about what is meant by "probabilistic", but there is a simple argument based on the Cauchy inequality (no spectral theory involved) which provides a so-called "ratio limit theorem" for return probabilities. It is valid for **any** reversible chain with an infinite stationary measure, i.e., any rev... | 2 | https://mathoverflow.net/users/8588 | 62986 | 38,937 |
https://mathoverflow.net/questions/62985 | 5 | Let $G$ be a compact connected Lie group, $T$ be some maximal torus in $G$ (that is, inclusion-maximal connected abelian subgroup). Then the union of tori $gTg^{-1}$, $g\in G$, is the whole $G$. This is well-known (4.21 in Adams book). My question is rather methodological: is there any proof without use of algebraic to... | https://mathoverflow.net/users/4312 | maximal tori cover compact Lie group | There are proofs that avoid algebraic topology: see for example Chapter 16 in Bump's *Lie Groups* or IV.5 in Knapp's *Lie Groups Beyond an Introduction*.
| 3 | https://mathoverflow.net/users/430 | 62989 | 38,940 |
https://mathoverflow.net/questions/62950 | 4 | What is known and not known about rational points of modular curves? What are some good references?
| https://mathoverflow.net/users/14674 | Rational points of modular curves | Dear Dick,
I am going to interpret *rational points* to mean points over $\mathbb Q$.
Given this,
Mazur's *Eisenstein ideal* paper, his *rational isogenies* paper, and various surveys
he wrote around that time (mid-to-late 1970s) give a good description (and also prove
the most of the key results).
Ogg also wrot... | 11 | https://mathoverflow.net/users/2874 | 62990 | 38,941 |
https://mathoverflow.net/questions/62949 | 15 | Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
My question is: is there a high-brow explanation for why positive definiteness and Fourier transforms go hand-in-hand... | https://mathoverflow.net/users/934 | Positive-Definite Functions and Fourier Transforms | Perhaps the phenomenon you are asking about is: why is the definition of a positive-definite function natural?
One answer is that positive-definite functions are exactly coefficients of group representations, in the following sense. If $\pi : \mathbb{R}\to U(H)$ is a unitary representation of $\mathbb{R}$ on some Hi... | 24 | https://mathoverflow.net/users/12660 | 62991 | 38,942 |
https://mathoverflow.net/questions/62977 | 3 | Every Mersenne number of index $n >2$
`$$
M_n = 2^n-1
$$`
is represented by the quadratic polynomial
$$
Q(x,y) = 28x^2+4y^2+28x+4y+7
$$
e.g.,
`$$
M_3=Q(0,0), M_4=Q(0,1), M_5=Q(0,2),M_6=Q(1,0),
$$
$$
M_7=Q(0,5),M_8=Q(2,4),M_9=Q(3,6),M_{10}=Q(1,15).
$$`
Since it is believed that there are an infinity of Mersenne p... | https://mathoverflow.net/users/11016 | Mersenne numbers represented by a Quadratic polynomial in two variables | I think this was proved by Iwaniec (for any quadratic in two variables satisfying the obvious necessary conditions).
| -1 | https://mathoverflow.net/users/2290 | 62992 | 38,943 |
https://mathoverflow.net/questions/62969 | 16 | We can derive from the Poisson summation formula the modularity of the Theta function, which results in the functional equation. In his book on the Riemann Zeta function, Patterson mentions also that there is a way back. How can this be done?
Equivalently, how can the modularity of the Theta function (which is Poiss... | https://mathoverflow.net/users/10400 | Why is the functional equation of the Riemann zeta function equivalent to the Poisson summation formula? | The theorem you are looking for is Theorem 10.2.17 in Henri Cohen's book Number Theory: Analytic and Modern Tools ([Google Books Link](http://books.google.com/books?id=HSCWL8SfJPQC&pg=PA177&dq=Generalized+poisson+summation+formula+theta+function&hl=en&ei=ZzC2TY6uKMHGgAeOlux4&sa=X&oi=book_result&ct=result&resnum=3&ved=0... | 10 | https://mathoverflow.net/users/12337 | 62999 | 38,945 |
https://mathoverflow.net/questions/62919 | 6 | Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.
In particular, here are the two questions I'm trying to answer right now:
Question 1: what is $\mathbb{Q}/\mathbb{Z}\_p$ tensored wit... | https://mathoverflow.net/users/14220 | reference for p-local and p-complete integers | I'm going to switch to a more standard notation, where the localization is $\mathbb{Z}\_{(p)}$ and the completion is $\mathbb{Z}\_p$.
For your first question, the general rule that concerns us is that the tensor product of a $p$-divisible group with a $p$-torsion group is zero. The reason is that if $a$ is a $p^n$th ... | 8 | https://mathoverflow.net/users/121 | 63005 | 38,950 |
https://mathoverflow.net/questions/62993 | 4 | I am looking for the preprint
>
> **A. Bondal,** Noncommutative
> deformations and Poisson brackets on
> projective spaces. **Preprint MPI/93-67**
>
>
>
which I could not find online. Does anyone have an eletronic version of it ?
I am interested in the conjecture made there which predicts that the locus wh... | https://mathoverflow.net/users/605 | Holomorphic Poisson brackets on Fano manifolds | Preprint of Bondal can be found here:
<http://www.mi.ras.ru/~akuznet/math/Bondal%20Non-commutative%20deformations%20and%20Poisson%20brackets%20on%20projective%20spaces.pdf>
Another related reference is the following paper
Fano threefolds with sections in $Ω^1(1)$, by Priska Jahnke and Ivo Radloff.
| 4 | https://mathoverflow.net/users/4428 | 63008 | 38,953 |
https://mathoverflow.net/questions/56145 | 6 | The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V , d)$ where ... | https://mathoverflow.net/users/6986 | Extension of the formality theorem? | I think this does only work for so-called homological vector fields, i.e. vector fields of degree 1 which self-commute. Then you have $[v,v]=0$, which is the Maurer-Cartan equation in $T\_{poly}$, and which insures that the corresponding derivative $d$ squares to zero.
Dealing with a Maurer-Cartan element you can si... | 2 | https://mathoverflow.net/users/7031 | 63014 | 38,959 |
https://mathoverflow.net/questions/62968 | 8 | Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all automorphic representations? Can they separate the cuspidal representation? What about the rest? Both the local and global picture ... | https://mathoverflow.net/users/10400 | Why is the Arthur trace formula so powerful? | I am not sure if I understood your question well. I try to give an answer to the following,
"Why is it expected that the trace formula implies cases of Langlands functoriality?"
hoping that it is the right question.
Very roughly I think the idea is as follows.
Let $G, H$ be two connected reductive groups, and ... | 7 | https://mathoverflow.net/users/4398 | 63018 | 38,961 |
https://mathoverflow.net/questions/63017 | 1 | If $\sigma(n)$ is the sum of the divisors of $n$, show that $\sigma(6q + 5) \equiv 0 (mod 6)$ for all positive $q$. Is this an instance of a more general rule?
| https://mathoverflow.net/users/14690 | Sigma Congruency | The older reference I found is a review of D.H. Lehmer on a paper of Gupta:
MR0014365 (7,273f)
Gupta, Hansraj
Congruence properties of $\sigma(n)$.
Math. Student 13, (1945). 25–29.
10.0X
Let $\sigma\_a(n)$ denote, as usual, the sum of the $a$th powers of the divisors of $n$.
Ramanathan has noted [Math. Student 11,... | 2 | https://mathoverflow.net/users/11016 | 63022 | 38,964 |
https://mathoverflow.net/questions/63021 | 2 | Let $f:X\to Y$ be a finite dominant morphism of nonsingular varieties over some algebraically closed field. Let $P\in X$ be a point of codimension one, then $f(P)$ is also of codimension one. We can define the ramification index $e\_P$ of $P$ in the same way one would do it if $X$ and $Y$ were curves, my main reference... | https://mathoverflow.net/users/9947 | When is the degree of the pull-back of a Weil divisor a constant multiple of its degree? | In the situation you consider the degree $\mathrm{deg}(f)$ is equal to the degree $[K(X):K(Y)]$ of the extension of the function fields of $X$ and $Y$. The precise requirements are that $X$ and $Y$ are normal varieties and that $f$ is finite. The base field is not required to be algebraically closed.
For varieties of... | 4 | https://mathoverflow.net/users/3556 | 63025 | 38,966 |
https://mathoverflow.net/questions/63027 | 6 | What are the matrices that you can write in the form $X \odot X^{-T}$, for a complex square matrix $X$, where $X^{-T}$ is the inverse of the *complex transpose* (not conjugate) and $\odot$ is the Hadamard (component-by-component) product?
In the $2\times 2$ case, you get the group of matrices in the form $$\begin{bma... | https://mathoverflow.net/users/1898 | Matrices that are Hadamard products of $X$ and $X^{-T}$ | There are some properties of this product in Horn and Johnson, "Topics in Matrix Analysis", Cambridge Univ Press 1991.
| 3 | https://mathoverflow.net/users/14695 | 63028 | 38,967 |
https://mathoverflow.net/questions/62859 | 42 | Given any reasonable formal system F (e.g., Peano Arithmetic or ZFC), we all know that one can construct a Turing machine that runs forever iff F is consistent, by enumerating the theorems of F and halting if it ever proves 0=1.
However, what interests me here is that the "obvious" such Turing machine will be an extr... | https://mathoverflow.net/users/2575 | "Simpler" statements equivalent to Con(PA) or Con(ZFC)? | The discussion in the comments has helped clarify your question for me. I believe that it is closely related to the following [remark by Harvey Friedman](http://www.cs.nyu.edu/pipermail/fom/2004-March/008003.html):
>
> I am convinced that trying to take consistency statements like Con(ZFC +
> measurable cardinals)... | 17 | https://mathoverflow.net/users/3106 | 63031 | 38,969 |
https://mathoverflow.net/questions/62996 | 14 | It is a standard fact that for any finite morphism of proper Noetherian $A$-schemes ($A$ being Noetherian), the pullback of an ample line bundle is ample. The usual proof of this fact is via Serre's cohomological criterion for ampleness. However, since the statement seems, on its face, to have nothing to do with cohomo... | https://mathoverflow.net/users/5094 | Non-cohomological proof that the pullback of an ample bundle by a finite morphism is ample | Unfortuatelly, this is too long for a comment.
Can't we directly show that for every coherent sheaf $F$ on $X$ we have that $F\otimes (f^\*L)^m$ is generated by global section for $m\gg 0$?
Since $f$ is finite and $L$ is ample, we have that $f\_\*F\otimes L^m$ is generated by global sections for $m\gg 0$. So there ... | 27 | https://mathoverflow.net/users/5537 | 63053 | 38,986 |
https://mathoverflow.net/questions/63064 | 20 | This question arised when I was trying to use [this answer](https://mathoverflow.net/questions/54311/determinant-and-exact-sequences-of-sheaves/54321#54321) to understand Reid's ["Young Person's guide to Canonical Singularities"](http://math.unice.fr/~sb/SpringSchool2009/YPG.pdf). In particular page 352 when computing ... | https://mathoverflow.net/users/1887 | Elementary short exact sequence of sheaves | Expanding the comment of Donu Arapura, let $X$ be a variety and $Y\subset X$ a subvariety.
Then, you have a short exact sequence of sheaves
$$
0\to\mathcal I\_Y\to\mathcal O\_X\to\mathcal O\_X/\mathcal I\_Y\to 0,
$$
where $\mathcal I\_Y$ is the ideal sheaf of $Y$. By definition, $\mathcal O\_X/\mathcal I\_Y=\mathcal O\... | 31 | https://mathoverflow.net/users/9871 | 63071 | 38,997 |
https://mathoverflow.net/questions/63080 | 1 | Hello,
If I have a matrix A, is it possible to construct a positive definite matrix M with the same range as range(A')? I am trying to use the property x'Mx > 0 to remove an absolute value constraint in an optimization problem.
Thanks
| https://mathoverflow.net/users/14710 | constructing a positive definite basis | If $M$ is a symmetric and positive definite matrix of size $n$ by $n$, then the range of $M$ is $R^{n}$. Unless $A^{T}$ also has $R^{n}$ as its range, you can't make this happen.
You'd have a much better chance of getting a useful answer if you asked your original question about dealing with the absolute value const... | 1 | https://mathoverflow.net/users/9022 | 63083 | 39,006 |
https://mathoverflow.net/questions/63087 | 7 | Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$:
Under "no overlap" constraints, I sequentially deposit circles of radii $r\_c$ on this surface, where the center-point of each circle is allowed real number coordinates and is chosen with uniform probability. To a... | https://mathoverflow.net/users/14324 | Packing density of randomly deposited circles on a plane | The model you describe seems to fall under what's called **"Random Sequential Addition"** or **"Random Sequential Adsorption"** in the literature; it's viewed as a higher dimensional analogue of the car parking problem. An early review in the physics literature on this type of model by J W Evans is [here](https://doi.o... | 9 | https://mathoverflow.net/users/353 | 63092 | 39,011 |
https://mathoverflow.net/questions/63098 | 0 | Given a series of samples $y\_1, y\_2, \ldots, y\_n \sim N(0,\Sigma)$, I'm looking to find the expection, $E(y^T y)$. It's fairly easy to show that $y^T \Sigma^{-1} y \sim \chi^2$, but then I get stuck. Maybe there isn't a closed form solution for $E(y^T y)$?
Thanks
Edited: Removed expectation around product.
| https://mathoverflow.net/users/14711 | Expectation of the sum of squares of a covariant normal samples | There isn't a closed form solution for the distribution, but there is for the expectation. It's called a Quadratic Form: <http://en.wikipedia.org/wiki/Quadratic_form_(statistics)>
$E(y^T y) = tr(\Sigma)$
Also, I think you mean to say that $y^T \Sigma^{-1} y \sim \chi ^2$, not $E(y^T \Sigma^{-1} y) \sim \chi ^2 $
| 3 | https://mathoverflow.net/users/14593 | 63101 | 39,015 |
https://mathoverflow.net/questions/63096 | 6 | Suppose you have a map $g:\Sigma \rightarrow G$ from a Riemann surface $\Sigma$ to a compact Lie group $G$. What is the obstruction to finding a $3$-manifold $W$, such that $\partial W = \Sigma$, and an extension of $g$ to a map $\tilde{g}:W\rightarrow G$? In the paper I'm reading they say it lies in $H\_2(G,\mathbb{Z}... | https://mathoverflow.net/users/13132 | Extending maps on Riemann surfaces | This has nothing to do with Lie groups. Let $X$ be any space, let $S$ be a closed, orientable surface and let $f : S \rightarrow X$ be a map. We then get a canonical homology class $f\_{\ast}([S]) \in H\_2(X;\mathbb{Z})$. If there exists a closed orientable $3$-manifold $M$ with boundary $S$ such that $f$ extends over ... | 11 | https://mathoverflow.net/users/317 | 63103 | 39,017 |
https://mathoverflow.net/questions/63094 | 6 | Q1: Do we have a criterion which allows us to say when is a profinite group $G$ topologically finitely generated?
For example, if $G$ is topologically finitely generated then, for a fixed integer $N$,
there are only finitely many open subgroups $H\leq G$ such that $[G:H]=N$. In general, I guess that this condition is... | https://mathoverflow.net/users/11765 | Criteria for topologically finitely generated profinite groups | See the theorem of Mann quoted by Jaikin in [this question](https://mathoverflow.net/questions/63029/finitely-generated-galois-groups) for a nice criterion guaranteeing topological finite generation in terms of the number of open subgroups of index N. See Pete Clark's answer to that question for pointers towards verify... | 6 | https://mathoverflow.net/users/431 | 63104 | 39,018 |
https://mathoverflow.net/questions/63056 | 22 | This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights.
Call a vector $v$ in $\mathbb{R}^2$ 'short' if it has modulus less than 1. Let $v\_1,\dots,v\_6$ be short vectors such that $\s... | https://mathoverflow.net/users/11727 | An elementary problem in Euclidean geometry | Lemma 1:
If the angle, $\theta$, between two 'short' vectors, $v\_1$ and $v\_2$ (placed head to tail) satisfies $\theta \leq \pi/3$, then their sum is a short vector.
Proof:
Place $v\_1$ at the origin, then the terminal point of $v\_1$ lies within the unit ball. Placing the $v\_2$ at the tail of $v\_1$ creates an... | 5 | https://mathoverflow.net/users/9015 | 63113 | 39,022 |
https://mathoverflow.net/questions/63107 | 3 | Can matrix representations of clifford algebras of type Cl(0,n) be found? Specifically for even orders
| https://mathoverflow.net/users/14713 | On matrix representations of the Clifford algebras of type $Cl(0,n)$ | The answer is **Yes**.
It follows from the classification of real Clifford algebras. First of all, write $n = 8p + q$, where $q=0,\dots,7$. Bott periodicity says that $Cl(0,n) \cong Cl(0,q) \otimes\_{\mathbb{R}} \mathbb{R}(16p)$, where $\mathbb{F}(N)$ denotes the real associative algebra of $N\times N$ matrices with ... | 6 | https://mathoverflow.net/users/394 | 63116 | 39,023 |
https://mathoverflow.net/questions/63118 | 6 | Let $p$ be a fixed prime number.
Question 1: Given a finite extension $K$ of $\mathbb{Q}\_p$ is there a totally real extension $F$ of $\mathbb{Q}$ and a place $v$ of $F$ over $p$ such that $F\_v = K$?
This is used in the proof of the local Langlands conjecture (thus I am quite sure that the answer is Yes) but I hav... | https://mathoverflow.net/users/13302 | How locally ubiquitous are totally real fields? | The answer to the first question is "yes". See [this paper of the Dokchitser brothers](http://arxiv.org/abs/0906.1815), Lemma 3.1 for the case where $K/\mathbb{Q}\_p$ is Galois. In the general case, apply the result to the Galois closure $K'$ of $K$ to get $F'$, identify the Galois group of the local fields with a deco... | 6 | https://mathoverflow.net/users/35416 | 63120 | 39,026 |
https://mathoverflow.net/questions/63122 | 3 | A Noetherian (commutative) ring $A$ is called *universally catenary* if every $A$-algebra of finite type is catenary. If one wants to know whether $A$ is universally catenary, then this definition suggests checking every $A$-algebra of finite type. Catenarity being preserved under taking quotients it clearly suffices t... | https://mathoverflow.net/users/11025 | Universal catenarity and Laurent algebras | Yes. Indeed, $S:=\mathrm{Spec}A[X]$ is covered by the open subspaces $U:=\mathrm{Spec}A[X,X^{-1}]$ and $V:=\mathrm{Spec}A[X,(X-1)^{-1}]$. Now $U$ is catenary by assumption, and $V$ is isomorphic to $U$, hence catenary. It follows easily that $S$ is catenary: just observe that if $Y\subset Z$ are irreducible and closed ... | 3 | https://mathoverflow.net/users/7666 | 63139 | 39,038 |
https://mathoverflow.net/questions/63109 | 1 | Dear all,
in order to prove the validity of my Galerkin approach of a certain variational problem, I need to check the so-called approximability property. In my case, it boils down to showing that for all $w\in L^2(\Omega)$, $\lim\_{h\rightarrow 0}\inf\_{w^h\in V^h}||w-w^h||=0$, where $\Omega=[0; 1]^d$, and $V^h$ is th... | https://mathoverflow.net/users/14712 | Approximation in $L^2$ by piecewise constant functions | A few word about the setting of $L^2$ approximation scheme by simple functions. If $(\Omega,\mathcal{A},\mu)$ is a measure space and $\mathcal{B}$ is a finite sub-algebra of $\mathcal{A}$, generated by the partition $\mathcal{E}$, the class $L^2(\Omega,\mathcal{B},\mu)$ is a finite dimensional subspace of $L^2(\Omega,\... | 4 | https://mathoverflow.net/users/6101 | 63140 | 39,039 |
https://mathoverflow.net/questions/63095 | 80 | I tried to answer an earlier question as to uses of GRR, just from my reading, although i do not understand GRR. Today i tried to understand the possible idea behind GRR. After editing my answer accordingly, it occurred to me i was asking a question instead of giving an answer. My question is roughly whether the follow... | https://mathoverflow.net/users/9449 | how does one understand GRR? (Grothendieck Riemann Roch) | Here is how I think about G-R-R in the context of moduli of curves. I realize now that I wrote something quite long.
Let me recall first the definition of the tautological ring. As a consequence of the results on the birational geometry of $\overline M\_g$ that there is no hope of understanding the whole Chow ring o... | 32 | https://mathoverflow.net/users/1310 | 63145 | 39,041 |
https://mathoverflow.net/questions/62508 | 3 | I have a question about eigenvalue problem on the geodesic ball in $n$-dimensional sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}$.
Consider the eigenvalue problem in the geodesic ball $\Omega=\{x\_{n+1}\geq c\}$ where $c\geq 0$:
$\Delta u+nu=0$ in $\Omega$
$u=0$ on $\partial\Omega$.
For the upper hemisphere, i.e... | https://mathoverflow.net/users/14579 | eigenvalue problem on the geodesic ball of sphere | You have the following estimate (see (3.10) of *Stability of minimal surfaces and eigenvalues of the Laplacian*, Barbosa & do Carmo).
Let $D$ a simply connected domain of $S^2$,
if $\vert D\vert \leq 2\pi$ then $\lambda\_1\geq \frac{4\pi}{\vert D\vert}$,
if $2\pi \leq \vert D\vert\leq 4\pi$ then $\lambda\_1\leq ... | 3 | https://mathoverflow.net/users/14694 | 63148 | 39,044 |
https://mathoverflow.net/questions/63134 | 5 | **Note:** This question was asked in stats.stackexchange.com and [math.stackexchange.com](https://math.stackexchange.com/questions/33617/a-sequence-of-order-statistics-from-an-iid-sequence), with expired bounties on both sites.
Given a sequence of iid random variables $X\_i$ (without loss of generality from $U(0,1)$... | https://mathoverflow.net/users/8652 | A sequence of order statistics from an iid sequence | Another way to describe this sequence of random vectors is that you have an unordered set of $k$ points, initially sampled independently, and at each step you add a point and then with probability $p$ remove the minimum, and with probability $1-p$ remove the maximum.
Given $x \in (0,1)$, the number of points in the $... | 5 | https://mathoverflow.net/users/2954 | 63152 | 39,046 |
https://mathoverflow.net/questions/63142 | 35 | The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a *Frobenius complement* if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$. Given such a Frobenius complement, the corresponding *Frobenius kernel* is defined by
$$
N = \left(G\backslash\bigc... | https://mathoverflow.net/users/35416 | Character-free proof that Frobenius kernel is a normal subgroup? | Nothing much to say here. There is (as of now) no proof of this fact without character theory. Although I think there is a direct counting proof when $H$ has even order, and a transfer argument
tells you that in a minimal counterexample, $H$ must be perfect (since $H$ is a Hall subgroup
of $G$). Hence in a minimal coun... | 26 | https://mathoverflow.net/users/14450 | 63156 | 39,049 |
https://mathoverflow.net/questions/63171 | 5 | I am currently writing a proof in which I need to use the fact that if a tree has no involutions, its automorphism group is trivial (ie, if a tree has any non-trivial automorphisms, then it has at least one automorphism of order 2). Equivalently, the order of the automorphism group of a tree is either equal to 1 or it ... | https://mathoverflow.net/users/14719 | Involution-free Trees are Asymmetric: Reference request | Look for "Automorphisms of Graphs" by P. J. Cameron (google finds the pdf), and look at the result of Polya on page 8 -- it states that the class of automorphism groups of trees is the smallest class containing the trivial group and closed under the operations of "direct product" and "wreath product with the symmetric ... | 6 | https://mathoverflow.net/users/11142 | 63176 | 39,059 |
https://mathoverflow.net/questions/63160 | 15 | Let $\phi,\psi\in\Pi\_1^0$ be independent of PA.
Is the disjunction $\phi\vee\psi$ independent of PA?
| https://mathoverflow.net/users/9833 | The disjunction property in Peano Arithmetic? | The answer is no, and here is a counterexample. The proof
relies on the double fixed point lemma, a generalization of
the usual Goedel fixed point lemma producing two statements
forming a fixed point with respect to a system, and I
provide a proof below. Using it, we may produce two
distinct sentences $\phi$ and $\psi$... | 20 | https://mathoverflow.net/users/1946 | 63183 | 39,063 |
https://mathoverflow.net/questions/63187 | 3 | Is there a result in the spirit of Bertrand-Chebyshev which talks about the existence of prime powers between n and 2n (or 3n or something like that) for n large?
| https://mathoverflow.net/users/2209 | prime powers between n and 2n | It follows from the prime number theorem that for fixed $k$, provided $n$ is sufficiently large, there is a prime $p$ such that $p^k$ is between $n$ and $2n$. Does this answer your question?
| 7 | https://mathoverflow.net/users/14302 | 63191 | 39,068 |
https://mathoverflow.net/questions/63188 | 5 | This is a question I was given in the exam: show that if $\lambda, \kappa$ are uncountable cardinals then
$$(H\_\lambda,\in) \prec\_1 (H\_\kappa,\in)$$
where $H\_\lambda$ is the class of all sets $x$ such that $|TC(x)| < \lambda$. ($TC(x)$ refers to the transitive closure of $x$.)
Equivalently, given any $\Sigma\_1$ ... | https://mathoverflow.net/users/14725 | $\Sigma_1$ elementary substructure | I assume that you mean to assume $\lambda\leq\kappa$.
For any $\vec x\in H\_\lambda$, we may by the Lowenheim-Skolem theorem find an elementary substructure $X\subset H\_\kappa$ with $TC(\vec x)\subset X$ and $X$ having size less than $\lambda$. In particular, $X$ has a witness $z$ for your $\Sigma\_1$ statement $\e... | 5 | https://mathoverflow.net/users/1946 | 63195 | 39,071 |
https://mathoverflow.net/questions/63196 | 4 | I am looking for lists of small
-solvable groups
-nilpotents groups
-simple groups.
By this I mean, do there exist lists of all the groups of order smaller than $n$, for $n$ reasonably big, satisfying one of the above properties?
Does anyone know a reference for this?
Thanks in advance.
| https://mathoverflow.net/users/12039 | Lists of small groups | In <http://en.wikipedia.org/wiki/List_of_finite_simple_groups> there is a list of simple groups up to 9828. Moreover, the GAP small group library contains all groups of order less than 2000.
| 7 | https://mathoverflow.net/users/10194 | 63198 | 39,072 |
https://mathoverflow.net/questions/63149 | 3 | Let $k$ be a field, $p,q$ positive integers, and let $R$ be the space of $(p \times q)$-matrices over $k$, and $S$ be the space of $(q \times p)$-matrices over $k$. For every matrix $A \in R$, we define a so-called *principal inner ideal*
$$[A] := \{ ABA \mid B \in S \} \subseteq R.$$
Define the *principal rank* of... | https://mathoverflow.net/users/12858 | ABA-product of matrices and length of chains of principal inner ideals | This is a nice exercise in linear algebra. It's more elegant to work without coordinates here, that is, let $R= \hom(U,V)$ and $S=\hom(V,U)$, where $U$ and $V$ are vector spaces over $k$. For $\alpha\colon U\to V$, show that
$$ [\alpha]:=\{ \alpha\beta\alpha \mid \beta\in S \}
= \{ \phi\colon U\to V \mid \ker\alpha \l... | 4 | https://mathoverflow.net/users/10266 | 63211 | 39,081 |
https://mathoverflow.net/questions/63186 | 7 | Given a topological space $M$ with a locally free $S^1$ action on it, assume the slice representation holds,(this is often the case, e.g. M is a smooth manifold) then this will make $M$ a principal $S^1$-orbibundle over $M/S^1$. What's the proper way to define the Euler class of this $S^1$-orbibundle? How should the na... | https://mathoverflow.net/users/14727 | Euler class of S^1-orbibundle | Here is a topological construction of such a class, in singular cohomology with rational coefficients.
Let $M$ be an $S^1$-space. Then there is the Borel construction $M // S^1 := ES^1 \times\_{S^1} M$. It comes with a map $f$ to $BS^1= CP^{\infty}$ which is a fibre bundle with fibre $M$.
Moreover, there is a map $q:... | 6 | https://mathoverflow.net/users/9928 | 63223 | 39,087 |
https://mathoverflow.net/questions/63129 | 3 | I have $f(x,y) = \frac{1}{2} (1 - x^2 - y^2)$, which is *a paraboloid centered around the origin* ([plot](http://www.wolframalpha.com/input/?i=plot+%281+-+x%5E2+-+y%5E2%29+%2F+2+from+x%3D-1+to+1+y%3D-1+to+1)).
Now I want to calculate the [solid angle](http://en.wikipedia.org/wiki/Solid_angle) (with the origin as the ... | https://mathoverflow.net/users/14716 | How do you calculate the solid angle of a rectangular, axis aligned section of a surface defined by a two dimensional function? | You need to include the differential surface area in your parametrized version of the integral. In effect, you replace the $\hat{n} dS$ term with $$\frac{\vec{f}\_x\times\vec{f}\_y}{\|\vec{f}\_x\times\vec{f}\_y\|} {\|\vec{f}\_x\times\vec{f}\_y\|} dxdy.$$
Although, really the two norms just cancel, so you needn't calc... | 2 | https://mathoverflow.net/users/14288 | 63231 | 39,091 |
https://mathoverflow.net/questions/63124 | 6 | I am currently reading some of Mackey's work on unitary representation.
Given a locally compact group $G$ and a unitary representation $\pi : G\rightarrow U(H)$. As far as I understood it, the representation $\pi$ is primary, if the von Neumann algebra generated by $\pi(g)$ for all $g \in G$ is a factor, see <http://... | https://mathoverflow.net/users/10400 | What is the difference between a primary representation and a irreducible representation? | I would recommend reading parts of Jacques Dixmier's book: "$C^\ast$-algebras" (North Holland, 1977 - translated from the french version of 1969), especially Chapters 5 (irreducible and factor representations of $C^\ast$-algebras) and Chapter 13 (the analogue for locally compact groups).
The regular representation of... | 4 | https://mathoverflow.net/users/14497 | 63234 | 39,093 |
https://mathoverflow.net/questions/62619 | 9 | It's known that
$$S\_2={2\over\pi} = {\sqrt{2}\over 2}{\sqrt{2+\sqrt{2}}\over 2}{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\dots$$
The terms in the product approaches 1, the same holds for the following convergent series, with $\phi$ the golden ratio
$$S\_1 = {\sqrt{1}\over\phi}{\sqrt{1+\sqrt{1}}\over\phi}{\sqrt{1+\sq... | https://mathoverflow.net/users/14597 | Generalized Vieta-product | I doubt there exists a closed formula for $n\ne 2$. In the case $n=2$ such formula exists only thanks to the double-angle formula for cosine.
Let $n$ be fixed and $c=c\_n$. Notice that $n=c^2-c$ and $c\to\infty$ as soon as $n\to\infty$.
Denote by $p\_k$ the $k$-th multiplier in the product $S\_n$. It can be easily ... | 6 | https://mathoverflow.net/users/7076 | 63235 | 39,094 |
https://mathoverflow.net/questions/63222 | 1 | Hi!
Does anyone know of any statement relating the probability that the second largest eigenvalue of a random graph is bigger than $x$ to the parameter $p$ in the $G\_{np}$ model?
| https://mathoverflow.net/users/44243 | PR[$\lambda_2 > x$] in $G_{np}$ model | Provided $pN$ is large in comparison to $N^{2/3}$ it looks like [this paper](http://arxiv.org/abs/1103.3869) by László Erdős, Antti Knowles, Horng-Tzer Yau, and Jun Yin has what you need. Here's the abstract:
>
> We consider the ensemble of adjacency matrices of Erdős–Rényi random graphs, i.e.\ graphs on $N$ vertic... | 1 | https://mathoverflow.net/users/353 | 63238 | 39,096 |
https://mathoverflow.net/questions/63233 | 3 | Let $\Phi\_n(t)$ be the $n$-th cyclotomic polynomial over the rationals. Stephens proved in 1971
that $\Phi\_p(q)$ and $\Phi\_q(p)$ are not always coprime for primes $p,q$ since one has
$$
gcd(\Phi\_p(q),\Phi\_q(p)) = 2pq+1
$$
for
$$
p=17, q=3313
$$
See the wiki page about the [Feit-Thompson Conjecture](https:/... | https://mathoverflow.net/users/11016 | Variant of Stephens result $\gcd(\Phi_p(q),\Phi_q(p))=2pq+1$ for $p=17$, $q=3313$ | You may want to look at Karl Dilcher and Joshua Knauer, On a conjecture of Feit and Thompson, in the book, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, 169–178, Fields Inst. Commun., 41, Amer. Math. Soc., Providence, RI, 2004, MR2076245 (2005c:11003). In addition to a t... | 1 | https://mathoverflow.net/users/3684 | 63243 | 39,101 |
https://mathoverflow.net/questions/63254 | 4 | Let $\langle L\_\alpha \rangle$ denote the hierarchy of constructible sets namely
$$L\_0 = \emptyset$$
$$L\_{\alpha+1} = \text{def}(L\_\alpha)$$
$$L\_{\gamma} = \bigcup\_{\alpha<\gamma}{L\_\alpha}$$
for $\gamma$ being limit ordinals
and
$$L = \bigcup\_{\alpha \in \text{Ord}}{L\_\alpha}$$
be the Godel constructible uni... | https://mathoverflow.net/users/14725 | Constructible universe within constructible universe | Yes. The construction relativizes level by level.
This can be verified by a straightforward induction. The point is that if $M$ is a transitive model of ZF and $D\in M$ is transitive, then ${\rm def}(D)\subset M$ and (therefore) ${\rm def}(D)={\rm def}(D)^M$.
To see this, either use that definability is $\Delta\_... | 5 | https://mathoverflow.net/users/6085 | 63256 | 39,111 |
https://mathoverflow.net/questions/63265 | 13 | When defining a functor (between categories), I am usually told that it assigns to each object of the source category an object of the target category. I do not find this very satisfactory since we are dealing with proper classes here. Judging by the definition, it must be possible to have the concept of a "map" betwee... | https://mathoverflow.net/users/9947 | What are "maps" between proper classes? | <http://en.wikipedia.org/wiki/Ordered_pair#Morse_definition>
Definition:
A relation $R$ is functional if and only if for all ordered pairs $\langle x,z\rangle$ and $\langle y,w\rangle$ in $R$, if $x=y$ then $z=w$.
Definition: If $R$ is a relation, $\operatorname{Range}(R) = \{y : (\exists x)(\langle x,y\rangle ... | 4 | https://mathoverflow.net/users/nan | 63270 | 39,119 |
https://mathoverflow.net/questions/63278 | 12 | Does computing the rank of an integer matrix have complexity polynomial in the size of the input?
The Gaussian elimination algorithm is polynomial in the number of elementary operations (addition and multiplication), but the intermediate size of of the matrix entries may go up exponentially. Are there other algorithm... | https://mathoverflow.net/users/1855 | Complexity of computing matrix rank over integers | Gaussian elimination *is* a polynomial-time algorithm. While it may not be obvious on the first sight, it can be implemented so that the intermediate entries have only polynomial size (bit length), because they happen to be equal to determinants of certain submatrices of the original matrix (or ratios thereof, dependin... | 12 | https://mathoverflow.net/users/12705 | 63285 | 39,127 |
https://mathoverflow.net/questions/63300 | 13 | The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime.
My question was inspired from Exercise 1.8 of Atiyah-Macdonald's book in Commutative Algebra:
*The set of prime ideals of a nonzero ring has minim... | https://mathoverflow.net/users/2720 | Is every poset the poset of prime ideals of a ring? | Hochster answered this question in his thesis (by finding such conditions). See: Hochster, M. [Prime ideal structure in commutative rings](https://doi.org/10.1090/S0002-9947-1969-0251026-X). Trans. Amer. Math. Soc. 142 1969 43–60.
| 15 | https://mathoverflow.net/users/14757 | 63303 | 39,137 |
https://mathoverflow.net/questions/63291 | 1 | Hello, I am looking for some words to describe what going on here. I'm sure this is not an original thought, so I'd like to read up on more from others who have thought out this topic further.
**FORMAT**
The problem is stated in a square matrix, consider the rows and columns having identical labels (labels are shown ... | https://mathoverflow.net/users/14753 | Dimensionality of a map -- distance | Testing whether $n$ points "need" $n-1$ dimensions (in the way you're referring to) using only their distances is done via Cayley-Menger determinants : <http://mathworld.wolfram.com/Cayley-MengerDeterminant.html>
More generally, you might want to take a look at this : <http://en.wikipedia.org/wiki/Distance_geometry>
... | 0 | https://mathoverflow.net/users/6325 | 63305 | 39,139 |
https://mathoverflow.net/questions/13167 | 53 | In Lecture Notes in Mathematics 1488, Lawvere writes the introduction to the Proceedings for a 1990 conference in Como.
In this article, Lawvere, the inventor of Toposes and Algebraic Theories, discusses two ancient philosophical "categories": that of BEING and that of BECOMING. And he's serious. While some of the m... | https://mathoverflow.net/users/2811 | Lawvere's "Some thoughts on the future of category theory." | The notion of a "category of Being" that Lawvere discusses there is the notion that more recently he has been calling a *category of cohesion* . I'll try to illuminate a bit what's going on .
I'll restrict to the case that the category is a topos and say *cohesive topos* for short. This is a topos that satisfies a sm... | 46 | https://mathoverflow.net/users/381 | 63309 | 39,142 |
https://mathoverflow.net/questions/63310 | 4 | This is (probably) an easy one:
Given a positive definite matrix $M$, find the positive definite matrix $X$, which minimizes $\textrm{tr}(X M)$ subject to $\det(X) = 1$.
Looking for how to find X, either a formula, an algorithm, or a reference to a paper.
-Thanks
| https://mathoverflow.net/users/14424 | Matrix optimization problem | If the eigenvalues of $M$ are $\lambda\_1$, $\lambda\_2$, ..., $\lambda\_n$, then the minimum is $n (\lambda\_1 \cdots \lambda\_n)^{1/n}$.
**Proof:** Your problem is invariant under conjugating $X$ and $M$ by a unitary matrix. So we may assume that $M$ is diagonal, with diagonal entries $\lambda\_1$, ..., $\lambda\_... | 10 | https://mathoverflow.net/users/297 | 63312 | 39,143 |
https://mathoverflow.net/questions/63329 | 5 | I'm not sure I have a lot more to say than the title. Let $G$ be your favorite simple algebraic group over $\mathbb{C}$, and let $$\overline {\mathrm{Gr}}\_\lambda= \overline{G(\mathbb{C}[[t]])\cdot t^\lambda \cdot G(\mathbb{C}[[t]])}/ G(\mathbb{C}[[t]]).$$ It's a commonly cited theorem that $\overline {\mathrm{Gr}}\_\... | https://mathoverflow.net/users/66 | What is the Picard group of a Schubert variety in the affine Grassmannian? | [From the comments] The proof of Proposition 13.2.19 in Kumar's "Kac-Moody groups, their flag varieties and representation theory" appears to provide the requested information.
| 6 | https://mathoverflow.net/users/425 | 63350 | 39,161 |
https://mathoverflow.net/questions/63348 | 3 | This question arises from a discussion with my friends on a commonly encountered IQ test questions: "What's the next number in this series 2,6,12,20,...". Here a "number" usually means an integer. I was wondering whether there is a systematical way to solve such problems.Let us call a point on a plane integer point if ... | https://mathoverflow.net/users/9858 | Integral interpolation by polynomial | Let the points be $(x\_j, y\_j), j=1\ldots n$. If the $x\_j$ are consecutive, the Lagrange interpolating polynomial will take integers to integers: easy proof by induction, using the difference operator $\Delta(p)(x) = p(x+1) - p(x)$. If not, choose arbitrary integers for the $y$ values to fill in the gaps.
| 4 | https://mathoverflow.net/users/13650 | 63356 | 39,166 |
https://mathoverflow.net/questions/63342 | 7 | I am looking for a reference for the following facts in functional analysis and topology. (If these "facts" are not true, I suppose I'm looking for the closest approximation which is true.)
Let $X$ be a locally compact Hausdorff topological space. Let $C(X)$ denote the ring of continuous complex-valued functions on $... | https://mathoverflow.net/users/3545 | Ring of continuous functions, reference request. | Let $X=Y\_1\sqcup Y\_2$, with both $Y\_i$ homeomorphic to $Y$. Then $C(X)=C(Y\_1)\oplus C(Y\_2)$. Given $a\in C(Y)$ let $\phi\colon a\mapsto a\oplus 0$, in the obvious way. This $\phi$ cannot be any $f^\*$, since $f^\*$ would necessarily map $1\mapsto 1\oplus 1$. I believe, this is a counter example to your putative th... | 2 | https://mathoverflow.net/users/2622 | 63360 | 39,169 |
https://mathoverflow.net/questions/62936 | 4 | Hi,
I'm currently wreading "Roe: *An Index Theorem on Open Manifolds, I*, J. Differential Geometry 27 (1988), p. 87-113" and there is a detail in the construction of the coarse index of a Dirac operator that I do not understand.
In Section 4 he explains the construction of abstract indices: Suppose we are given an ... | https://mathoverflow.net/users/13356 | A detail in the construction of the coarse index of a Dirac operator in "Roe: An Index Theorem on Open Manifold, I" | This is a "left and right" issue. The point is that if $A$ is an algebra, the operation of left multiplication by a fixed $a\in A$ (considered as a map $A\to A$) is linear as a map of right $A$-modules.
| 6 | https://mathoverflow.net/users/14768 | 63367 | 39,176 |
https://mathoverflow.net/questions/63301 | 30 | Let $f:X\rightarrow Y$ be a morphism of schemes.
1. When $PicY\rightarrow PicX$ is an embedding and $f\_{\*}\mathscr{O}\_{X}$ is invertible, it is the structure sheaf of $Y$.
2. In the proof of Zariski's Main Theorem, we have: If $f$ is birational, finite, integral, and $Y$ is normal, then $f\_{\*}\mathscr{O}\_{X}$ i... | https://mathoverflow.net/users/14008 | When will the pushforward of a structure sheaf still be a structure sheaf? | **Q**: Exactly what information is contained in $f\_\*\mathscr O\_X$? Look at the
definition. For any $U\subseteq Y$ open, $f\_\*\mathscr O\_X(U) = \mathscr O\_X(f^{-1}(U))$ =
regular functions on $f^{-1}(U)$. So the information in $f\_\*\mathscr O\_X$ is related
to the sets in $X$ of form $f^{-1}(U)$.
Cases where $f... | 24 | https://mathoverflow.net/users/9449 | 63374 | 39,181 |
https://mathoverflow.net/questions/63345 | 6 | I'm looking for "famous" or otherwise well-known 2d Riemannian manifolds which have non-constant curvatures but have a non-trivial Killing vector field. Of course there are tons of spaces like these, for instance if we parametrize the plane (or a subset of it) by $(r,\phi)$ then any conformal rescaling of the flat metr... | https://mathoverflow.net/users/4526 | "Famous" 2d Riemannian manifolds with non-constant curvature | A Killing field is preserved by the Ricci flow. By a theorem of Daskalopoulos, Hamilton and Sesum ([arXiv:0902.1158](http://arxiv.org/abs/0902.1158)), on a compact surface an ancient (defined for all negative time) solution to the Ricci flow which is not a shrinking soliton is diffeomorphism equivalent to the Fateev-On... | 11 | https://mathoverflow.net/users/9471 | 63382 | 39,185 |
https://mathoverflow.net/questions/63383 | 23 | I've read the claim that Fréchet spaces that are not Banach spaces never have a dual that is a Fréchet space, but have not been able to find a proof of this statement. Is it trivial or does someone have a reference?
| https://mathoverflow.net/users/1478 | Which Fréchet spaces have a dual that is a Fréchet space? |
>
> For any locally convex and metrizable space $E$, its strong dual is metrizable if and only if $E$ is normable.
>
>
>
This and related properties of (F)-spaces are discussed in detail in *Topological Vector Spaces I* by Köthe (see §29.1, pp. 393-394 in the English edition).
| 26 | https://mathoverflow.net/users/5371 | 63385 | 39,186 |
https://mathoverflow.net/questions/63319 | 1 | [Principal congruence subgroups of $SL(n, \mathbb{Z})$](https://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz)
gives a discription of automorphisms of $SL(n,\mathbb Z)$. Is it true for n even too?. Hau-Reiner's paper gives generators for the group of automorphisms of $SL(n,\mathbb Z)$ ... | https://mathoverflow.net/users/13835 | Principal congruence subgroups of SL(n,Z) | Yes, I think the statement is also true for $n >2$ even (see Guntram's answer of the question [Automorphisms of $SL\_n(\mathbb{Z})$](https://mathoverflow.net/questions/57235/automorphisms-of-sl-n-mathbbz)).
As a second reference a paper of O'Meara may serve. It can be found [here](http://www.digizeitschriften.de/dms... | 2 | https://mathoverflow.net/users/10194 | 63390 | 39,189 |
https://mathoverflow.net/questions/63280 | 1 | I was trying to understand the Mukai Fourier transform from his paper : Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math Journal, 1981.
I am not very familiar with Derived categories and even less familiar with $D^-(X)$ and so this question, as Mukai works with $D^-(X)$. My k... | https://mathoverflow.net/users/11395 | The Mukai Fourier Transform and Derived Categories | For geometry the most informative category is $D^b(coh X)$. However, not every functor preserves coherence and boundedness. For example, it is possible that $L\_if^\*(F) \ne 0$ for arbitrary large $i$ even if $F$ is a sheaf. This is why sometimes people have to work in $D^-(X)$ --- otherwise the derived pullback will n... | 1 | https://mathoverflow.net/users/4428 | 63396 | 39,194 |
https://mathoverflow.net/questions/62744 | 9 | Let $S$ be a scheme over a field $k$, and let $G$ be a reductive group scheme over $S$. Let us call it *trivial*, if it is a pull-back of a group scheme over $k$ via the structure morphism $S\to k$. Is it always true that $G$ becomes trivial after a certain etale base change $S'\to S$? I am willing to assume that $S$ i... | https://mathoverflow.net/users/6772 | Is every reductive group scheme etale locally trivial? | Reductive groups schemes over $S$ are classified by $H^1\_{fpqc}(S,Aut\_G)$, see SGA 3 Exp. XXIV.
| 6 | https://mathoverflow.net/users/5107 | 63401 | 39,196 |
https://mathoverflow.net/questions/63398 | 18 | This is a very naive question which really does little more than highlight my ignorance of how converse theorems really work.
Take an algebraic gadget which should be conjecturally associated to an automorphic representation. For example, take a finite image continuous complex representation of
the absolute Galois gr... | https://mathoverflow.net/users/1384 | Why isn't meromorphic continuation enough for converse theorems? | Dear Kevin,
My understanding of the current meromorphic continuation results is that they do something along the lines of expressing a given Galois representation as the induction of a virtual (i.e. positive and negative coefficients) combination of various automorphic Galois representations.
To get true automorphy... | 15 | https://mathoverflow.net/users/2874 | 63407 | 39,200 |
https://mathoverflow.net/questions/63205 | 42 | The title pretty much says it all. I am revisiting complex analysis for the first time since I "learned" some as an undergraduate. I am trying to wrap my head around why it should be the case that a function which is differentiable once should be differentiable twice. I know a proof (Use Cauchy integral formula and dif... | https://mathoverflow.net/users/1106 | Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions? | I recalled a comment of Ahlfors at the beginning of Chapter 4 of his book "Complex Analysis" which asserted the existence of such a proof, and it led me to the paper "A proof of the power series expansion without Cauchy's formula" by E. H. Connell and P. Porcelli (Bull. Amer. Math. Soc. 67 (1961), 177-181), where they ... | 20 | https://mathoverflow.net/users/6545 | 63408 | 39,201 |
https://mathoverflow.net/questions/44167 | 19 | I'd like to state explicitly a problem which was somehow hidden in [my three-week-old post](https://mathoverflow.net/questions/41660/is-the-operator-norm-always-attained-on-a-0-1-vector):
>
> Does there exist an absolute constant $c>0$ with the property that for any matrix $M\in{\mathcal M}\_{m\times n}(\{0,1\})$ (... | https://mathoverflow.net/users/9924 | Is the norm of a $0-1$ matrix (almost) attained on a $0-1$ vector? | As I have figured out recently, the answer is **no**. The full proof is somewhat technical and I cannot supply full details within the framework of an MO post, but here is the idea behind the construction.
Start with a symmetric matrix $A\in{\mathcal M}\_{n\times n}(\{0,1\})$ such that the Perron-Frobenius eigenvalue... | 11 | https://mathoverflow.net/users/9924 | 63417 | 39,204 |
https://mathoverflow.net/questions/63414 | 4 | The strong Whitney embedding theorem states that any smooth (Hausdorff and second-countable) manifold can be smoothly embedded in Euclidean space. John Nash went on to show that any Riemannian manifold could be embedded into Euclidean space in such a way that the metric of the manifold would coincide with the standard ... | https://mathoverflow.net/users/1648 | "Nash Style" Embedding Theorem for Connections | The standard connection is the Levi-Civita connection of the flat metric. So if you have an embedding such that the given connection is the (projection of the) flat connection then you can also induce a metric on the embedded manifold. Hence the given metric was already a Levi-Civita connection of a Riemannian metric o... | 9 | https://mathoverflow.net/users/12482 | 63419 | 39,206 |
https://mathoverflow.net/questions/63412 | 15 | Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ primes, which is asymptotically equivalent to $\frac{n^2}{2\log{n}}$. It should thus be possible to find estimates for $s... | https://mathoverflow.net/users/1737 | Upper bounds for the sum of primes up to $n$ | By partial summation
$$ s(n) = n\pi(n)-\sum\_{m=2}^{n-1}\pi(m) $$
so by the Prime Number Theorem
$$ s(n) = \frac{n^2}{\log n}-\sum\_{m=2}^{n-1}\frac{m}{\log m}+O\left(\frac{n^2}{\log^2 n}\right). $$
The sum on the right is
$$ \sum\_{m=2}^{n-1}\frac{m}{\log m} = \int\_2^n \frac{x}{\log x}dx + O\left(\frac{n}{\log n}\rig... | 25 | https://mathoverflow.net/users/11919 | 63427 | 39,212 |
https://mathoverflow.net/questions/63402 | 16 | Some months ago (October 2010), in the context of the [Workshop on Set Theory and the Philosophy of Mathematics](http://www.phil.upenn.edu/Workshop%20on%20Set%20Theory%20and%20the%20Philosophy%20of%20Mathematics/abstracts), Professor Donald A. Martin gave a talk entitled "Philosophical issues about the hierarchy of set... | https://mathoverflow.net/users/6466 | Martin's "Philosophical Issues about the Hierarchy of Sets" | Of course there are no universally agreed-upon answers to these philosophical questions, and if you are interested in Martin's views specifically, then I suggest that you read his articles. Meanwhile, allow me simply to explain a few of the issues arising in the specific questions you mention.
* "One cannot quantify ... | 12 | https://mathoverflow.net/users/1946 | 63432 | 39,215 |
https://mathoverflow.net/questions/63435 | 2 | Let $X$ be a finite set, $(X^{\mathbb Z}, T)$ is the shift, i.e. the Tikhonov topological space of all bi-infinite words in $X$, $T$ shifts the words one letter to the right. A subshift is a closed subset of $X^{\mathbb{Z}}$ stable under $T$.
Is there a recent survey about the problem of equivalence of subshifts?
... | https://mathoverflow.net/users/nan | Equivalent subshifts | The paper *[Open Problems in Symbolic Dynamics](http://www-users.math.umd.edu/~mmb/open/)* by Mike Boyle discusses the conjugacy problem for shifts of finite type and sofic shifts.
More details can be found in books such as *[An Introduction to Symbolic Dynamics
and Coding](http://www.math.washington.edu/SymbolicDyna... | 5 | https://mathoverflow.net/users/728 | 63436 | 39,217 |
https://mathoverflow.net/questions/63431 | 0 | This question is related to my previous question. Suppose that a group $G$ acts freely and properly on a Riemaniann manifold $(M, g)$. Than the orbits form a foliation for $M$. For $p \in M$, let $V\_p$ be the tangent space at $p$ of the leaf containing $p$, and let $V^\perp\_p$ be the orthogonal complement of $V\_p$. ... | https://mathoverflow.net/users/9584 | Does an abelian group acting on a riemaniann manifold define an othogonal foliation? | Any contact manifold with a weakly compatible metric provides a counterexample, where the action is $\mathbb{R}$ acting by the Reeb flow. "Weakly compatible" here just means that there is a metric $g$ such that if $V$ is the Reeb vector field and $\xi$ the contact distribution, $\xi \perp\_g V$.
| 1 | https://mathoverflow.net/users/2510 | 63453 | 39,226 |
https://mathoverflow.net/questions/63451 | 8 | Let $K$ be a field. Let $V/K$ be an affine variety in $A^m$. Let f be a polynomial map (and hence a "morphism of finite type") $f:V\to A^n$. A theorem of Chevalley's tells us that im(f) is either a variety or "almost" a variety - that is, im(f) is a variety $W$ with perhaps a few varieties of lower dimension cut out fr... | https://mathoverflow.net/users/398 | Degree of image of a polynomial map | Call $r$ the dimension of $W$, $d$ the degree of $V$, and $e$ the largest degree of one of the $f\_i$. Then I claim that $\deg W$ is at most $de^r$.
We may assume that $K$ is algebraically closed. We may also assume that the dimensions of $V$ and $W$ are the same (otherwise, cut $V$ with generic hyperplane sections u... | 6 | https://mathoverflow.net/users/4790 | 63463 | 39,232 |
https://mathoverflow.net/questions/63440 | 13 | The action of a group $G$ on a topological space $X$ can be viewed as a functor $F: G \to \mathcal{Top}$ with $F(\*)=X$. (Here I'm viewing a group as a category with one object, $ \* $, and the morphisms are isomorphisms labeled by the group elements.)
We can extend this idea and define the action of a groupoid $\mat... | https://mathoverflow.net/users/5000 | Groupoid actions on spaces | Perhaps the most natural example is given by universal covers?
Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X\_x$ be the universal covering of
$X$ taken at $x$ (the fiber at $y \in X$ is the homotopy classes of paths $[0,1]\to X$ which start
at $x$ and end at $y$, where we are taking homotopy classes ... | 26 | https://mathoverflow.net/users/8032 | 63469 | 39,235 |
https://mathoverflow.net/questions/63358 | 9 | This a repost of a question I asked at Stack Exchange:
<https://math.stackexchange.com/questions/35264/congruences-for-fermat-quotients>
I didn't get a complete answer to my question, so I'm trying again here.
If $p$ is a prime number and $a$ is relatively prime to $p$, then by Fermat's Little Theorem, the **Fer... | https://mathoverflow.net/users/5373 | Congruences between Fermat quotients | Let just try a look at your first result:
Corollary 1 in Johnson, Wells paper above says:
Let $\epsilon = 1,-1$ and let $p=2^r+ \epsilon$ be a prime number.
Then
$$
q\_p(2) \equiv \frac{\epsilon}{r} \pmod{p}
$$
Your result for a Mersenne prime (taking $\epsilon =-1$ above) follows from this:
Observe that (tr... | 3 | https://mathoverflow.net/users/11016 | 63474 | 39,238 |
https://mathoverflow.net/questions/63439 | 35 | Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL\_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL\_{k}(\mathbb{R})$ deformation-retracts onto $O\_k$, then $BGL\_{k}(\mathbb{R})\simeq BO\_k$, which is a cute way (though it's certainly overkill) of proving that eve... | https://mathoverflow.net/users/303 | How can we detect the existence of almost-complex structures? | **Edit:** Now updated to include reference and slightly more general result.
**Edit 2:** Includes remark about integrability.
Similar to Francesco Polizzi's answer, there is the following Theorem concerning 6-manifolds.
A closed oriented 6-dimensional manifold $X$ without 2-torsion in $H^3(X,\mathbb{Z})$ admits an... | 16 | https://mathoverflow.net/users/380 | 63476 | 39,239 |
https://mathoverflow.net/questions/63446 | 2 | I am reading materials about the determinant defined by Knudsen-Mumford
<http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=103495&vfpref=html&r=11&mx-pid=437541>
which assigns a graded line bundle to a perfect complex of locally free coherent $\mathcal{O}\_X$-modules. Here, a graded line bundle is just a... | https://mathoverflow.net/users/3849 | the inverse of determinant line bundle? | A **Picard groupoid** is a symmetric monoidal category $G$ where all morphisms are isomorphisms and such that for any object $x\in G$, the functor $x\otimes-\colon G\rightarrow G$ is an equivalence of categories. Graded line bundles form a Picard groupoid. An **inverse object** to $x\in G$ is an object $x^\star\in G$ t... | 3 | https://mathoverflow.net/users/12166 | 63482 | 39,242 |
https://mathoverflow.net/questions/63465 | 19 | My impression is that one of the celebrated results of class field theory the [principal ideal theorem](http://en.wikipedia.org/wiki/Principal_ideal_theorem) namely that given a number field $K$ and its maximum unramified abelian extension L, every ideal in the ring of integers $O\_K$ of $K$ becomes principal in the ri... | https://mathoverflow.net/users/683 | Where does the principal ideal theorem (from CFT) go? | Among the generalizations that I can recall off the top of my head are:
* the generalization to ray class groups already mentioned by Kevin, proved by Iyanaga pretty much immediately after
Furtwängler's proof;
* Furtwängler's own theorem saying that if the class group is an elementary
abelian $2$-group, then its b... | 15 | https://mathoverflow.net/users/3503 | 63512 | 39,262 |
https://mathoverflow.net/questions/63423 | 124 |
>
> Is there a chess position with a finite number of pieces on the infinite chess board $\mathbb{Z}^2$ such that White to move has a forced win, but Black can stave off mate for at least $n$ moves for every $n$?
>
>
>
This question is motivated by a question [posed here](https://mathoverflow.net/questions/27967... | https://mathoverflow.net/users/14302 | Checkmate in $\omega$ moves? | Here is my first try at a solution. Your idea was a good one, but
bishops are better than rooks, I surmise.
The two pictures here are placed in some distinct parts of the infinite board.
The first just ensures it is White to move (in check), and that White's king
will never play a role, as capturing a black unit, whi... | 60 | https://mathoverflow.net/users/5267 | 63517 | 39,263 |
https://mathoverflow.net/questions/63528 | 3 | Let $(X,B,\mu)$ and $(Y,C,\nu)$ be probability spaces, and let $m$ be the product measure. Let $f:X \times Y \rightarrow [0,\infty )$ be a $B \otimes C $ measurable function, $1 < p < \infty$, and define $f\_x(y):=f(x,y)$.
What is the most general condition on $f$ to make sure that: $||f||\_{L\_p(m)}=\int{||f\_x||\_{... | https://mathoverflow.net/users/14623 | Probability measure product space | yes, it is the only case. Denote $g(x)=(\int{|f(x,y)|^pd\nu(y))}^{1/p}$, then your condition is $||g||\_1=||g||\_p$ which is known to imply that $g$ is a constant function, see [here](http://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality).
| 8 | https://mathoverflow.net/users/8699 | 63533 | 39,271 |
https://mathoverflow.net/questions/63532 | 2 | It is known that, for $n \ge 3, 2 < p< 2^\*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. Let $G=O(n\_1) \times O(n\_2)\times\cdots\times O(n\_k)$, with
$n\_1+n\_2+\cdots+n\_k=n, n\_i \ge 2$, and $k \ge 1$. Define an action of $G$ on $H^1(\mathbb{R}^n)$ by $g.u=u\circ g^{-1}$, an... | https://mathoverflow.net/users/14361 | Sobolev imbedding | Of course yes, basically you achieve compactness with $H^1\_r$ because you have local regularity plus decay at infinity (pointwise decay like $|x|^{(1-n)/2}$ to be precise, by Strauss-type inequalities). If I'm not mistaken, any weighted $H^1$ space with norm $\|\langle x\rangle^\epsilon u\|\_{H^1}$ , $\epsilon>0$, sho... | 6 | https://mathoverflow.net/users/7294 | 63536 | 39,274 |
https://mathoverflow.net/questions/63490 | 21 | Let $x\_1,x\_2,…,x\_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma\_{ij})\_{1\leq i,j\leq n}$.
Let $m$ be the maximum of the random variables $x\_{i}$
$$
m=\max\{x\_i:i=1,2,\ldots,n\}
$$
What can one say about $m$? Can we at least compute its mean and variance?
More specifically... | https://mathoverflow.net/users/13825 | Maximum of Gaussian Random Variables | If the correlations decay fast enough $\sigma\_{ij}(n) = o(1/\log n)$, then the asymptotic distribution of the maximum is the same as if the variables were independent (i.e.
the standard Gumbel distribution) - see:
Limit Theorems for the Maximum Term in Stationary Sequences, S.M. Berman (Ann. Math. Statist. 1964)
[Li... | 15 | https://mathoverflow.net/users/1778 | 63537 | 39,275 |
https://mathoverflow.net/questions/63519 | 32 | (Sorry if this is too elementary for this site)
I’m having some trouble understanding sheaf cohomology. It’s supposed to provide a theory of cohomology “with local coefficient”, and allow easy comparison between different theories like singular, Cech, de Rham and Alexander Spanier. What I don’t understand is: what’s ... | https://mathoverflow.net/users/14800 | Coefficients in cohomology | This (elementary and perfectly standard) example might help show the power of sheaves with non-constant coefficients:
First, think about the circle $S^1$. Suppose you want to understand (real) line bundles on the circle. You can certainly cover the circle with two open contractible subsets $U\_1$ and $U\_2$ (which yo... | 33 | https://mathoverflow.net/users/10503 | 63539 | 39,277 |
https://mathoverflow.net/questions/63541 | 2 | Let $f:X\to Y$ be a finite, dominant morphism of projective varieties. I suspect that $\deg(f)\cdot\deg(Y)=\deg(X)$ always holds, where $\deg(f)=[K(X):K(Y)]$. If required, we may assume that $X$ and $Y$ are projective over an algebraically closed field (of characteristic zero even). My question is whether this is true ... | https://mathoverflow.net/users/9947 | Relating degree of projective varieties and a finite morphism between them | This is not true in general.
In fact, let $H\_X$ be a hyperplane section of $X$ and $H\_Y$ be a hyperplane section of $Y$, with respect to the fixed embeddings $X \subset \mathbb{P}^N$ and $Y \subset \mathbb{P}^M$. Then
$\deg X= (H\_X)^n, \quad \deg Y =(H\_Y)^n$,
where $n= \dim X = \dim Y$.
Now requiring
$\... | 5 | https://mathoverflow.net/users/7460 | 63542 | 39,279 |
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