parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/168040 | 11 | Is there an algorithm which on input "$(a,p)$" (where $0\leq a<p$ are integers) takes time polynomial in $\log p$ and outputs "NOT PRIME" if $p$ is not prime and otherwise outputs the Legendre symbol $(a/p)$?
By the [AKS primality test](http://en.wikipedia.org/wiki/AKS_primality_test), it suffices to assume $p$ is pr... | https://mathoverflow.net/users/29873 | Can the Legendre symbol be calculated in polynomial time? | For quadratic reciprocity, you need the [Jacobi symbol](http://en.wikipedia.org/wiki/Jacobi_symbol). It's an extension of the Legendre symbol to composite $p$ that still satisfies quadratic reciprocity, so you can just apply quadratic reciprocity to your heart's content without worrying about primality. The Jacobi symb... | 20 | https://mathoverflow.net/users/4720 | 168042 | 87,076 |
https://mathoverflow.net/questions/168047 | 1 | Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding
$$\phi: (M,\omega)\to (\mathbb CP^N, \omega\_{FS})$$ So $$\phi^\*\omega\_{FS}=\omega+\frac{i}{2\pi}\partial\bar \partial \epsilon $$
where $\epsilon$ is a smooth... | https://mathoverflow.net/users/nan | A question about Quantized closed Kaehler manifolds | I do not believe this is the case. If you have any smooth complex submanifold $X$ of $\mathbb{CP}^n$, then the Kahler form on $\mathbb{CP}^n$ pulls-back to a Kahler form $\omega$ on $X$ (so $\phi^\*\omega\_{FS}=\omega$). You can take $X$ to be any smooth projective variety. These are not all simply-connected. For insta... | 0 | https://mathoverflow.net/users/25358 | 168052 | 87,079 |
https://mathoverflow.net/questions/168057 | 1 | I am currently the convergence of different processes. Doing this, I ended up with this expression and was wondering whether it is true that$$\lim\_{n \rightarrow \infty}\sum\_{k=0}^{n} \frac{|(1-\frac{n p\_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0?$$ -For the case that $np\_n \rightarrow \lambda $ and $p\_n \rightarrow 0$.... | https://mathoverflow.net/users/nan | Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$? | I'd say it's true, by the dominated convergence theorem for series. You may think the $n$-th sum as a series whose terms $c\_{n,k} $ vanish for $k > n$. For fixed $k$, the coefficient $c\_{n,k}$ tends to $0$ as $n\to\infty$ (recall that if $z\_n\to z$ then $(1+z\_n/n)^n\to e^z$). Moreover $|c\_{n,k}|$ is uniformly boun... | 1 | https://mathoverflow.net/users/6101 | 168059 | 87,080 |
https://mathoverflow.net/questions/168039 | 5 | From *Counterexamples in Topology* by Steen and Seebach (2nd edition) example 129 page 145 we have an example of **connected and totally path-disconnected space**.
It is defined as follow:
Fix $p= (1/2,1/2)$. Let $C$ be the Cantor set in the unit interval $[0,1]$. Let $E \subset C$ be the subset of $C$ that is the en... | https://mathoverflow.net/users/3958 | Beyond Cantor's Teepee | I happened to run across an answer to 2. (and therefore 1.) just by clicking on one of the related MO links, finding the same question in a comment by Timothy Gowers under [MO16578](https://mathoverflow.net/questions/16578/can-a-connected-planar-compactum-minus-a-point-be-totally-disconnected) that was later answered i... | 10 | https://mathoverflow.net/users/2926 | 168060 | 87,081 |
https://mathoverflow.net/questions/168053 | 2 | From wikipedia:
>
> Shoenfield's absoluteness theorem shows that $\Pi^1\_2$ and $\Sigma^1\_2$
> sentences in the analytical hierarchy are absolute between a model $V$
> of ZF and the constructible universe $L$ of the model, when interpreted
> as statements about the natural numbers in each model.
>
>
>
**Qu... | https://mathoverflow.net/users/26080 | Is there an analogue of Shoenfield's absoluteness theorem, but for $\mathrm{On}$? | Perhaps the solution lies in the Levy absoluteness theorem.
We define the Levy hierarchy of formulas as follows.
1. $\Sigma\_0,\Pi\_0,\Delta\_0$ are formulas in the language of set theory which only have bounded quantifiers.
2. $\Sigma\_{n+1}$ formulas equivalent to formulas of the form $\exists x\varphi$, where $\... | 6 | https://mathoverflow.net/users/7206 | 168061 | 87,082 |
https://mathoverflow.net/questions/168065 | 0 | The idea is to build in ZFC using replacement, a set REPLACEMENT(x ∈ A: TERM(x))
from a set and a term in the same way the set {x ∈ A: FORMULA(x)} is built
using specification from a set and a formula.
Given a set A and a TERM(x) with a free variable x, can be proved, using
the axiom scheme of replacement, the exist... | https://mathoverflow.net/users/51212 | Possible no standard use of replacement axiom | If you're just asking whether this can be proved, the answer is yes (assuming that "For all SET(z)" means "For all sets z"). More importantly, this can be proved also for extensions by definitions of ZFC. That's important, because the official vocabulary of ZFC has no function symbols, so the only terms are variables; ... | 2 | https://mathoverflow.net/users/6794 | 168068 | 87,085 |
https://mathoverflow.net/questions/168045 | 5 | OEIS [A226181](http://oeis.org/A226181):
```
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...
```
Primes $p$ such that $p-1$ divided by the period of the binary expansion of $1/p$ equals $2^x$ for some nonnegative integer $x$. C... | https://mathoverflow.net/users/33646 | Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers? | Let's first show that the numbers in the sequence are Poulet numbers. It is known that the length of the period of the binary expansion of $\frac{1}{n}$ is $ord\_n 2$, the multiplicative order of $2 \pmod n$. So let $n$ be a composite number such that $\frac{n-1}{ord\_n 2}=2^x$. Then $n$ is odd. Therefore, $2^n\equiv 2... | 5 | https://mathoverflow.net/users/23008 | 168074 | 87,091 |
https://mathoverflow.net/questions/165103 | 1 | Suppose $X,Y$ are smooth varieties over $\mathbb{C}$, and let $K\_i \in D^b(X \times Y), i=1,2$ be objects in the derived category of bounded complex of coherent sheaves on $X \times Y$. Then there are Fourier-Mukai functors (FM functors) $\Phi\_{K\_i}$ from $D^{b}(X)$ to $D^b(Y)$ associated to kernel $K\_i$:
$$F \ma... | https://mathoverflow.net/users/29730 | Morphism between Fourier-Mukai functors implies the morphism between kernels? | This is somewhat sad, but I think (part of) what we've learned from the whole triangulated-vs-dg story is the following pseudo-statement: the bare category of functors Fun(D(X),D(Y)) is the wrong thing to take -- it should be replaced the category D(X x Y).
There is a nice example, due to the usual Bondal-Orlov-Bridg... | 4 | https://mathoverflow.net/users/46690 | 168081 | 87,094 |
https://mathoverflow.net/questions/168084 | 14 | The well-known Hahn-Mazurkiewicz theorem characterizes those nonempty Hausdorff spaces $X$ that admit a continuous surjection $\alpha: [0, 1] \to X$ from the closed unit interval: it is necessary and sufficient that $X$ be a compact, connected, locally connected metrizable space.
My Googling skills did not enable me... | https://mathoverflow.net/users/2926 | continuous images of open intervals | Call $A$ an HM-space if there is a continuous surjection $I\to A$ (where $I$ is the closed interval $[0,1]$). Note that if $A$ is an HM-space, then it is path-connected.
Theorem: If $X$ is path-connected, then the following are equivalent:
1. $X = \bigcup\_{n=1}^\infty A\_n$ where each $A\_n$ is an HM-space
2. ther... | 9 | https://mathoverflow.net/users/3634 | 168086 | 87,096 |
https://mathoverflow.net/questions/168077 | 3 | Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via a wick rotation, and the theories were constructed in euclidean space rather than Minkowski space.
Now, I have two quest... | https://mathoverflow.net/users/51216 | How does Constructive Quantum Field Theory work? | An euclidean theory which fulfills the Osterwalder-Schrader axioms gives a Wightman theory on Minkowski space via analytical continuation of the correlation function by the <http://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem>
E.g. a 4 dim theory gives a Wightman theory on 3+1 dim Minkowski space, so there is n... | 6 | https://mathoverflow.net/users/10718 | 168096 | 87,101 |
https://mathoverflow.net/questions/168093 | 7 | Have any of Neukirch's papers on anabelian geometry been translated? I'm mostly interested in:
[Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper](http://www.ams.org/mathscinet-getitem?mr=0244211) (1969)
[Kennzeichnung der endlich-algebraischen Zahlkörper durch die Galoisgruppe der maximal auf... | https://mathoverflow.net/users/43108 | Neukirch's papers and theorem | You can find this in the last chapter of [Neukirch-Schmidt-Wingberg, Cohomology of Number Fields], <http://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/index-de.html>.
See also [Ivanov, On a generalization of the Neukirch-Uchida theorem, arXiv:1309.3046, 2013] <http://www-m11.ma.tum.de/fileadmin/w00bnb/www/people/ivano... | 9 | https://mathoverflow.net/users/nan | 168097 | 87,102 |
https://mathoverflow.net/questions/123760 | 21 | This question is related to [question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?"](https://mathoverflow.net/questions/92206/) but is not exactly equivalent in my opinion. It is even suggested in one of the answers to [92206](https://mathoverflow.net/questions/92206/) that "th... | https://mathoverflow.net/users/20995 | Topological characterization of the closed interval $[0,1]$ | Consider the class of all Hausdorff compacts with distinct points (i.e. which have more than $1$ point) that are [absolute retracts](http://planetmath.org/absoluteretract) in the class
of Hausdorff compacts.
Then $[0,1]$ is up to homeomorphism the only member of this class that embeds into every other.
| 4 | https://mathoverflow.net/users/20995 | 168110 | 87,106 |
https://mathoverflow.net/questions/168109 | 6 | Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number
of prime numbers $p \leq n$ in the residue class $r$ (mod $m$).
Further let $1 = r\_1 < r\_2 < \dots < r\_{\varphi(m)} = m-1$ be the prime residues
(mod $m$), and let $\sigma \in {\rm Sym}(\{r\_1, \dots, r\_{\varphi(m)}\})$
be a permutation of... | https://mathoverflow.net/users/28104 | Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$) | The answer is conjectured to be "yes." Further, assuming GRH and GSH [Rubinstein and Sarnak, Chebyshev's Bias, Exp. Math 1994](http://projecteuclid.org/euclid.em/1048515870) even could show results on the logarithmic densities of the resepactive sets (which in particular is non-zero). I do not know about general explic... | 12 | https://mathoverflow.net/users/nan | 168111 | 87,107 |
https://mathoverflow.net/questions/168113 | 6 | I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type theory (themselves inextricably linked by the Curry–Howard–Lambek correspondence), but these rather seek an entirely diff... | https://mathoverflow.net/users/602 | Why can't mathematics be formalised in terms of classes rather than sets? | Putting together your two remarks
>
> just deal with the classes implicitly defined by predicates in some logical system.
>
>
>
and
>
> we could envisage a universe of discourse being something like the naturals (recursively defined), with only first-order classes.
>
>
>
seems to me to yield an inform... | 7 | https://mathoverflow.net/users/23141 | 168116 | 87,110 |
https://mathoverflow.net/questions/168114 | 2 | Let $K(x)$ be the complete elliptic integral of the first kind
(the argument is the parameter $m = k^2$).
Let $$ A = \int\_0^1 \arcsin(K(x)) dx$$
With precision $1000$ decimal digits $\Re A = \frac{\pi}{2}$.
>
> Is this true?
>
>
>
According to Wolfram Alpha for the indefinite integral
there is no result ... | https://mathoverflow.net/users/12481 | Conjectured closed form for definite integral | In the interval [0,1], K(x) takes real values greater than $\pi/2$. Hence $\arcsin(K(x))$ (with the appropriate choice of branch) is equal to $\pi/2$ plus something imaginary.
| 11 | https://mathoverflow.net/users/12120 | 168118 | 87,112 |
https://mathoverflow.net/questions/168108 | 1 | I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = f(2\pi),f'(0) = f'(2\pi)$. I know that this equation is called a Mathieu equation and it is well-studied in the literatu... | https://mathoverflow.net/users/nan | Spectrum of Mathieu equation | A good reference is Whittaker Watson, vol. 2.
| 1 | https://mathoverflow.net/users/25510 | 168129 | 87,114 |
https://mathoverflow.net/questions/168132 | 5 | A plane Poisson process with uniform intensity scatters "sites" about the plane. If I'm not mistaken, in a sense the "average" Voronoi diagram of that set of sites is a honeycomb. I know it's been proved that the average number of edges of the cells is $6$, and I've read (but not in anything published very recently) th... | https://mathoverflow.net/users/6316 | "Average" Voronoi diagrams without probability? | Here is an argument that Béla Bollobás showed me once. (this was motivated by a physics paper where a simulation was done showing that the average number of edges per face
was 5.997$\pm$ 0.005).
Take a large number of seeds (i.e. points generating the Voronoi diagram) and make the assumption that there are no *multi... | 10 | https://mathoverflow.net/users/11054 | 168134 | 87,116 |
https://mathoverflow.net/questions/165797 | 9 | In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group $\mathrm{Cl}(\mathcal O)$ of $\mathcal O$ is isomorphic to the quotient $I\_K(f) / P\_{K,\mathbb Z}(f)$, where $I\_K(f)$ is the fre... | https://mathoverflow.net/users/48499 | Class groups of orders | The answer is yes, essentially by weak approximation.
Let $\mathbb{I}\_K=\prod'\_{v}K\_v$ denote the finite Idelles over $K$. Now let $\prod\_v\mathcal{O}\_v^{\times}=\widehat{\mathcal{O}}^{\times}$ denote the multiplicative group of the completion of $\mathcal{O}$. Then (as Cox explains) you can write
$$Cl(\mathc... | 5 | https://mathoverflow.net/users/4181 | 168145 | 87,118 |
https://mathoverflow.net/questions/168153 | 3 | Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha\_1,\ldots,\alpha\_n,\beta\_1,\ldots,\beta\_n\}$, which are disjoint except for $\alpha\_i$ and $\beta\_i$ intersecting in a single point (relative orientations chosen... | https://mathoverflow.net/users/43462 | Does every canonical decomposition of the intersection form come from a canonical homology basis? | Yes, every such decomposition gives rise to a symplectic automorphism $h$ of $Z^{2n}$ (sending standard symplectic generators to the generators of $A$ and $B$ respectively).
Now, use the fact that the natural homomorphism from the mapping class group of $X$ to $Sp(2n,Z)$ is surjective.
| 6 | https://mathoverflow.net/users/21684 | 168162 | 87,125 |
https://mathoverflow.net/questions/168143 | 1 | I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal.
In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that guarantees me that there is a real eigenvalue? Or is there a way to prove the existence for all $n \in \mathbb{N}$? Of course ... | https://mathoverflow.net/users/nan | Existence of a real eigenvalue | It will be better if you write the definition of your matrix in a more readable way.
From what you wrote, it seems that your matrix satisfies $M(k,k+1)M(k+1,k)\geq 0$.
With this condition, all eigenvalues are real.
In general the eigenvalues of a Jacobi (3-diagonal) matrix will not change if you replace
both off diag... | 7 | https://mathoverflow.net/users/25510 | 168170 | 87,127 |
https://mathoverflow.net/questions/99587 | 7 | Let $f$ be a continuous real-valued function defined on an $n$ dimesional simplex $\Sigma\subset \mathbb{R}^n $. The classical simplicial approximation scheme provides a sequence $f\_k$ of piecewise affine functions converging uniformly to $f$: namely, on each simplex $S$ of the $k$-th baricentric subdivision of $\Sigm... | https://mathoverflow.net/users/6101 | $W^{1,1}$ simplicial approximation | You can represent on every simplex $\sigma$ of your triangulation the derivative $D f\_k$ by
$$
D f\_k = \frac{1}{\lvert \sigma \rvert} \sum\_{i = 0}^n \int\_{\sigma} \Bigl(\frac{1}{(1 - \beta\_i (x))^n} - 1\Bigr)\, Df(x)[a\_i - x]\,dx D\beta\_i,
$$
where $\beta\_i$ are the maps that give barycentric coordinates
$$
... | 4 | https://mathoverflow.net/users/42047 | 168176 | 87,130 |
https://mathoverflow.net/questions/168149 | 1 | If $Z\_1,Z\_2,Z\_3,\ldots$ are i.i.d. with $P(Z\_i=-1) = P(Z\_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum\_{i=1}^n Z\_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so that for any continuous bounded function $f,$ we have $\mathbb{E}f\left(\frac{\sum\_{i=1}^n Z\_i}{\sqrt{n}}\right)\t... | https://mathoverflow.net/users/7576 | Central limit theorem and convergence of means | The following result appears in Billingsley's book *Convergence of probability measures* (1968, and I guess the 1995 edition).
>
> **Theorem.** Let $(Y\_n)\_{n\geqslant 1}$ be a sequence of non-negative random variables defined on a probability space $(\Omega,\mathcal F,\mu)$ such that $Y\_n\to Y$ in distribution a... | 1 | https://mathoverflow.net/users/17118 | 168177 | 87,131 |
https://mathoverflow.net/questions/168173 | 3 | It's a consequence of Lebesgue's theorem that every measurable $E\subset\mathbb{R}^n$ has a metric density that's $1$ a.e. on $E$ and $0$ a.e. on $\mathbb{R}^n\setminus E$. What are the most general conditions on the measure space for this property to hold?
| https://mathoverflow.net/users/12518 | Metric density theorem in most general setting? | It is true for doubling metric-measure spaces.
Check Theorem 1.8 in "Lectures on analysis on metric spaces" by Heinonen.
| 4 | https://mathoverflow.net/users/1441 | 168181 | 87,133 |
https://mathoverflow.net/questions/168174 | 6 | We have the Adams SS with
$$ E\_2^{p,q} = Ext^{p,q} \_{E^\*(E)}([S,E],[S,E]) $$
where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.
I was wondering if there is a SS for arbitrary compactly generated triangulated categories of which this is special case.
More specifically I am curious... | https://mathoverflow.net/users/29548 | Adams Spectral Sequence for Triangulated Categories | In addition to the notes of Haynes Miller see <http://jdc.math.uwo.ca/papers/ideals.pdf>
| 7 | https://mathoverflow.net/users/16785 | 168191 | 87,137 |
https://mathoverflow.net/questions/168146 | 9 | Let me give some motivation, which also explains how I arrived at the question. We may let the finite group $G$ act on itself by conjugation, and this makes the group ring into a $\mathbb{Z}G$-module which affords character $ \theta = \sum\_{\chi \in {\rm Irr}(G)}(\chi \overline{\chi}).$ It follows that for each irredu... | https://mathoverflow.net/users/14450 | When is the conjugation character almost multiplicity free? | Extraspecial $2$-groups and $S\_3$ are the only nonabelian examples.
We may write $\theta$ as a sum of the permutation characters on the orbits, which in our case are the conjugacy classes: In notation of the question,
$$ \theta = \sum\_{i=1}^k (1\_{C\_G(x\_i)})^G .$$
Suppose that $x$ and $y$ are not conjugate. I... | 8 | https://mathoverflow.net/users/10266 | 168199 | 87,141 |
https://mathoverflow.net/questions/156540 | 0 | This is the opposite question to this one: [Example of locally convex space such that its weak and initial topology coincide](https://mathoverflow.net/q/156538/13356).
If we have a normed vector space $X$ than its norm topology and weak topology coincide if and only if $X$ is finite dimensional. Now I'm interested in... | https://mathoverflow.net/users/13356 | Sufficient condition such that weak and initial topology coincide for a locally convex space | The following is mentioned in the book by [H.Schaefer](http://www.goodreads.com/book/show/2302578.Topological_Vector_Spaces) (IV, Exercise 6(a)), but with a little bit different formulation:
**Theorem.** A complete locally convex space $X$ over $\mathbb K$ ($\mathbb K=\mathbb R$ or $\mathbb C$) has weak topology (i.e... | 2 | https://mathoverflow.net/users/18943 | 168200 | 87,142 |
https://mathoverflow.net/questions/168036 | 6 | Summary:
--------
Given $X$ (the discrete Fourier transform of some unknown vector $x$ of length $N$), is there any shortcut to computing $X'$ (the Fourier transform of $x$ after padding it with $N$ zeros)?
Of course, this can be accomplished by computing $x$ from $X$ (via the inverse discrete Fourier transform), t... | https://mathoverflow.net/users/17087 | Shortcut from discrete Fourier transform F{x} to zero-padded F{x:0...0} | 
There is no local relation between the odd and even index components of $X'$ (the even index components might be nearly all zero, and the odd index components all nonzero, see figure from these [lecture notes](http://users.ece.gatech.edu/~mcclella/courses/ee4078/... | 2 | https://mathoverflow.net/users/11260 | 168207 | 87,144 |
https://mathoverflow.net/questions/168178 | 7 | I am quite sure the following fact must have been known for set theorists, though I could not find it anywhere.
If $r$ is random over $L$ and $x\in L[r]\setminus L$, then there must be some real $r\_0$ random over $L$ so that $r\_0\in L[x]$ and $x\in L[r\_0]$.
So where I could find it?
**Addition**
Monroe Eskew... | https://mathoverflow.net/users/14340 | Demuth's theorem in set theory | I'm not sure if this specific claim is stated explicitly anywhere, but it follows from the more general discussion about intermediate extensions in Jech, page 247. If we take any $x \in L[r]$, then there is a complete subalgebra $A$ of the random algebra associated to $\dot{x}$, with the property that $L[x]=L[G\_r \cap... | 9 | https://mathoverflow.net/users/11145 | 168211 | 87,145 |
https://mathoverflow.net/questions/168201 | 13 | Let $A$ be a $2\times 2$ matrix which we assume to be *contracting*, i.e., the exists $\alpha\in(0,1)$ such that
$$
\|A {\mathbf x}\|\_2\le \alpha\|{\mathbf x}\|\_2,\quad \forall {\mathbf x}\in\mathbb R^2.
$$
Let for $i=1,2$,
$$
T\_i(\mathbf x)=A\mathbf x+\mathbf v\_i
$$
for some vectors $\mathbf v\_1, \mathbf v\_2$.... | https://mathoverflow.net/users/8131 | Dimensions of self-affine sets | I'm going to expand on Gerald's answer. Indeed, there is no known general formula for the Hausdorff and/or box counting dimension of self-affine sets, even in your situation which is a priori the simplest possible (ambient dimension $2$, only two maps, and strong separation).
Falconer's classical theorem from "The Ha... | 8 | https://mathoverflow.net/users/11009 | 168214 | 87,147 |
https://mathoverflow.net/questions/168208 | 0 | Let $G$ be a locally compact group which acts non-singularly on a standard probability space $(X,\mu)$. Consider the Koopman representation $\pi\_X:G\rightarrow U(L^2(X,\mu))$ defined by
$(\pi\_X(g)\xi)(x)=\xi(g^{-1}x)\sqrt{\chi(g^{-1},x)}$ for $g\in G$, $\xi\in L^2(X,\mu)$.
We say that the action is amenable if the ... | https://mathoverflow.net/users/9401 | Amenability of the Koopman representation | Yes, see, for instance the original paper by Bekka (1990).
| 2 | https://mathoverflow.net/users/8588 | 168216 | 87,148 |
https://mathoverflow.net/questions/168219 | 1 | Let $X$ be the blow-up of $\mathbb{P}^n$ at $k$ general points. We can assume $k\leq n+4$. Let
$$D = aH-b\_1E\_1-...-b\_kE\_k$$
be a divisor on $X$. Are there conditions on $a,b\_1,...,b\_k$ ensuring that $D$ is ample? I just know results of this kind for $n = 2,3$. What about higher dimension?
In particular is the ... | https://mathoverflow.net/users/nan | Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points | If $k\leq n+3$ and if $n = 3, k = 7$, or $n = 4, k = 8$ then $X$ is a Mori Dream Space. You can find this here <http://arxiv.org/abs/math/0505337>.
In particular this implies that the cone of curves $NE(X)$ is polyhedral and generated by the finitely many classes of curves. Therefore a divisor $D$ in $X$ is ample if ... | 2 | https://mathoverflow.net/users/14514 | 168223 | 87,149 |
https://mathoverflow.net/questions/168192 | 4 | We have all been there, when a formula works for the first 30 parameters,
but it is not sufficient for a proof. My question is where one can actually just check a finite number of cases, to conclude that a formula is correct.
That is, let $f,g : S \to \mathbb{N}$ be two functions from some (infinite) set $S$,
say the... | https://mathoverflow.net/users/1056 | Results where complexity bounds implies finite number of test cases | I think that most of the answers to this [MO question](https://mathoverflow.net/questions/112097/important-open-problems-that-have-already-been-reduced-to-a-finite-but-infeasibl/112131#112131) are examples that you are looking for.
I'll rehash my own answer there in your language so that you can see what I mean.
Let... | 2 | https://mathoverflow.net/users/2233 | 168229 | 87,151 |
https://mathoverflow.net/questions/168203 | 1 | Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X\_1,\dots,X\_n)$ a polynomial in $k[X\_1,\dots,X\_n]$.
I am looking for notes, books or surveys detailing the link between the $S$-integers solutions of equations of the type $F(X\_1,\dots,X\_n)=c$ and quasi-$S$-integral ... | https://mathoverflow.net/users/51272 | Link between integral points on varieties and solutions to Diophantine equations | As Felipe mentions, I discuss this a bit in the listed paper. There's also a reasonable discussion, I believe, in Vojta's Springer Lecture Note. But since, as you say, many (maybe even most) of the relations are easy, people don't tend to include the details in papers, and even in a book they'd likely be exercises. The... | 7 | https://mathoverflow.net/users/11926 | 168232 | 87,153 |
https://mathoverflow.net/questions/168180 | 6 | I was reading [This question on MO](https://mathoverflow.net/questions/36175/is-there-any-criteria-for-whether-the-automorphism-group-of-g-is-homomorphic-to) and suddenly faced another question.
Is there any necessary and sufficient condition for $G$ such that there exist a group $G'$ that $G \cong Aut(G')$ ?
What ca... | https://mathoverflow.net/users/51263 | Is there any criterion for whether the group G is isomorphic to the automorphism group of another group? | Let me expand on my comment. There are two papers that you should look at:
>
> Iyer, Hariharan K. *On solving the equation $Aut(X)=G$*.
> Rocky Mountain J. Math. 9 (1979), no. 4, 653–670.
>
>
>
The scope of this paper is when your group $G'$ is **finite**. The MathSciNet review gives a perfect precis:
> ... | 9 | https://mathoverflow.net/users/801 | 168236 | 87,154 |
https://mathoverflow.net/questions/168221 | 0 | Given a sequence of functions $f\_k \in L^1([0,1])$ such that $||f\_k||\_{L^1(0,1)}\leq C$.
Is there a subsequence $\{k\_l\,|\,l\in \mathbb N\}\subseteq \mathbb{N}$ such that for $\mathcal{L}^1$-almost every $x\in (0,1)$ it holds: $\limsup\_{k\_l\in \mathbb N} |f\_{k\_l}(x)| < \infty$?
Although the statement is qu... | https://mathoverflow.net/users/51280 | Uniform boundedness in $L^1[0,1]$ implies finite $\limsup$ almost everywhere for a subsequence? | In general, this is not possible. I'll use the following fact:
Given $\delta, \epsilon >0$, we can find subsets $E\_n\subset [0,1]$ with $|E\_n|<\delta$ so that $\left|\bigcup E\_{n\_j} \right| > 1-\epsilon$ for any selection of infinitely many of these sets.
We can do this explicitly, as follows: Let $E\_1=[0,\del... | 1 | https://mathoverflow.net/users/48839 | 168237 | 87,155 |
https://mathoverflow.net/questions/168224 | 3 | I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any single place to learn about Hamiltonian groups.
I have found some books (even available online from the author!) that come... | https://mathoverflow.net/users/51275 | Recommended textbooks for Hamiltonian group actions? | The best Book about Hamiltonian action is
[Moment Maps, Cobordisms, and Hamiltonian Group Actions
Par Victor Guillemin,Yael Karshon,Viktor L. Ginzburg](http://lib.org.by/get/M_Mathematics/MD_Geometry%20and%20topology/MDdg_Differential%20geometry/Ginzburg%20V.L.,%20Guillemin%20V.,%20Karshon%20Y.%20Moment%20maps,%20co... | 2 | https://mathoverflow.net/users/nan | 168238 | 87,156 |
https://mathoverflow.net/questions/168089 | 2 | Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum\_{i=1}^{m} (A\_i + S\_i) \le L < \sum\_{i=1}^{m+1} (A\_i + S\_i) \big)$$
where $A\_i \sim \text{exp}(\lambda), S\_i \sim \text{exp}(\mu), L \sim \text{exp}(\lambda)$ are mutually independent, $\lambda \neq \mu$ are two positive integer... | https://mathoverflow.net/users/28199 | How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$? | Call the left resp. right hand sum $R\_m$ resp. $R\_{m+1}$. As $L$ is $\exp(\lambda)$ and independent of $(R\_m,R\_{m+1})$ , taking expection with resp. to $L$ first gives
$$\mathbb{P}(R\_m\leq L < R\_{m+1})=\mathbb{E}(e^{-\lambda R\_m}-e^{-\lambda R\_{m+1}})= \left(\dfrac{1}{2}\dfrac{\mu}{\mu+\lambda}\right)^m\left(1-... | 3 | https://mathoverflow.net/users/48831 | 168240 | 87,158 |
https://mathoverflow.net/questions/168222 | 4 | Assume $\kappa$ is uncountable and $\phi$ is an $L\_{\infty,\kappa}$ sentence. Let $K$ be the collection of models of $\phi$ partially ordered by $\prec\_{\infty,\kappa}$. It is well-known that $K$ is closed under unions of increasing chains of length $\kappa^+$, but the argument fails for increasing chains of length $... | https://mathoverflow.net/users/13694 | Counterexample for closedness under union of $\prec_{\infty,\kappa}$ chains | In $L\_{\infty,\kappa}$, you can express that a linear order has cofinality at least $\kappa$, since you can say that every less-than-size-$\kappa$ family is bounded. But having cofinality at least $\kappa$ is not generally preserved under unions of elementary chains of length $\lt\kappa$, although it would be for long... | 4 | https://mathoverflow.net/users/1946 | 168259 | 87,165 |
https://mathoverflow.net/questions/162695 | 2 | The general Degree Sequence Problem asks for a simple undirected graph (that is a graph without self-loops and with no more than one edge between any pair of nodes) for which it holds that the degrees of the nodes are $D = (d\_1, d\_2, ..., d\_n)$.
From the Erdős–Gallai theorem it follows that such a graph can be fou... | https://mathoverflow.net/users/46958 | Degree Sequence Problem on $k$-Partite Graphs | The problem a hand can be modeled as a restricted degree sequence problem as explained in that paper: <http://arxiv.org/pdf/1301.7523v3.pdf>
Or as a Degree Sequence Problem with Associated Costs as in <http://research.microsoft.com/en-us/um/people/nvishno/site/publications_files/mvdeg02.pdf>
Unfortunately, both pro... | 1 | https://mathoverflow.net/users/46958 | 168289 | 87,170 |
https://mathoverflow.net/questions/168264 | 9 | In May and Sigurdsson's *Parameterized Homotopy Theory*, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow D)$, with induced maps $C\overset{g}\to D$ and $C\overset{i}\to A$, there are natural isomorphisms:
$$(1)~~j^\ast f... | https://mathoverflow.net/users/11546 | Shriek push-forward for parameterized spectra | Yes, the [Beck-Chevalley condition](http://ncatlab.org/nlab/show/Beck-Chevalley%20condition) holds for [parameterized spectra](http://ncatlab.org/nlab/show/parametrized+spectrum). A proof appears as prop.4.3.3 in
* Michael Hopkins, Jacob Lurie, *Ambidexterity in K(n)-Local Stable Homotopy Theory* ([pdf](http://www.m... | 9 | https://mathoverflow.net/users/381 | 168297 | 87,172 |
https://mathoverflow.net/questions/168282 | 7 | This question was first asked on MathSE but nobody answered.
In his proof of Lemma A2.5 in his book *Commutative Algebra with a View towards Algebraic Geometry*, Prof. Eisenbud writes something like this:
Let R be a commutative ring, $M$ an $R$-module, $S(M)$ the symmetric algebra (the quotient of the tensor algebr... | https://mathoverflow.net/users/3333 | Problem with Eisenbud's Lemma "Symmetry of Diagonalization"? | I think this is a mistake in Eisenbud's book.
It seems, however, that Eisenbud only uses Lemma A2.5 in two places: in the proof of Proposition A2.4, and in the proof of Proposition-Definition A2.6.
**DISCLAIMER:** All that I am saying below is meant to refer to the case when $M$ is an even $R$-module. I am not maki... | 7 | https://mathoverflow.net/users/2530 | 168299 | 87,173 |
https://mathoverflow.net/questions/168292 | 8 | I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.
Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed permutation matrices of order $2^nn!$. This is proven in an article by Walter Feit which appeared as "Orders of finite... | https://mathoverflow.net/users/48084 | Maximal order of finite subgroups of $GL(n,Z)$ | In fact, I have a copy of a preprint by Feit of this paper. I have not checked the results, but here is what Feit says: the group of signed permutation matrices is of maximal order as a finite subgroup of ${\rm GL}(n,\mathbb{Q})$, except in the following cases (Feit's Theorem A).
$n = 2, W(G\_{2})$ of order $12$.
... | 12 | https://mathoverflow.net/users/14450 | 168305 | 87,175 |
https://mathoverflow.net/questions/165418 | 0 | Let $R$ be a commutative ring and $\{M\_i\}\_{i\in I}$ be a family of $R$-modules. We know that there are functors $^\widetilde{\ }: R{\rm -Mod}\to {\rm Qco(Spec}R)$ and $\Gamma({\rm Spec}R, -): {\rm Qco}({\rm Spec}R) \to R{\rm -Mod}$ which give us an equivalence of the category of $R$-modules and quasi-coherent $R$-mo... | https://mathoverflow.net/users/38585 | operatiing quasi-coherator functor on a product of modules | Let $P = \prod\_{i \in I}^{\mathsf{Mod}(\mathrm{Spec}(R))} \widetilde{M\_i}$ and $Q = \prod\_{i \in I}^{\mathsf{Qcoh}(\mathrm{Spec}(R))} \widetilde{M\_i}$. The superscript indicates in which category the product is taken.
Then we have $Q = \widetilde{\prod\_{i \in I} M\_i}$, since $M \mapsto \widetilde{M}$ is an equi... | 2 | https://mathoverflow.net/users/2841 | 168312 | 87,178 |
https://mathoverflow.net/questions/168313 | 0 | Can you provide me a counter example for this.
Suppose that I have a sequence of probability measures
$(\mu\_{r,t})\_{r,t>0}$ on a compact space metric $X.$
Suppose additionally that:
there exists a probability measure
$\mu\_{r,0}$ such that $\mu\_{r,t}\to^{\*,t\to 0}\mu\_{r,0}$
and
there exists a probabili... | https://mathoverflow.net/users/39115 | A very natural question in weak* topology | Let $X=\{1,2\}$ and define $\mu\_{r,t}:= \delta\_1$ either if $r \geq t$, or if both $r<t$ and $\lfloor \frac{1}{r}\rfloor$ is even. Define $\mu\_{r,t}:=\delta\_2$ if both $r<t$ and $\lfloor \frac{1}{r}\rfloor$ is odd. Clearly $\lim\_{t \to 0}=\delta\_1$ for every $r>0$, and $\lim\_{r \to 0} \lim\_{t \to 0}\mu\_{r,t}=\... | 4 | https://mathoverflow.net/users/1840 | 168322 | 87,180 |
https://mathoverflow.net/questions/168205 | 3 | **Theorem 2** [1, p.46] Let $X$ be a non-singular projective algebraic variety of dimension $n$, and $D$ a numerically effective $\mathbb{Q}$-divisor such that $(D^n)>0$. We assume that the support of the non-integral part $D-[D]$ is a divisor with normal crossing on $X$. Then $H^i(X,O\_X([-D]))=0$ for $i<n$.
**Proof... | https://mathoverflow.net/users/nan | A question about kawamata's proof of vanishing for big and nef $\mathbb{Q}$ divisors | ACL already basically answered this in the comments (including what to do in local coordinates), but maybe this is worth explaining in a bit more detail and in more generality.
Whenever one has a finite map between varieties $f : X' \to X$ we have an induced map of fraction fields $K(X) \subseteq K(X')$. Then $K(X')$... | 3 | https://mathoverflow.net/users/3521 | 168329 | 87,182 |
https://mathoverflow.net/questions/168328 | 2 | The title is pretty self explanatory: I'm looking for an english translation of Delignes inventiones paper "La conjecture de Weil pour les surfaces K3."
Anyone know if such a thing exists?
Thanks!
| https://mathoverflow.net/users/4181 | Is there an english translation of Delignes "La conjecture de Weil pour les surfaces K3."? | Yes there is. Just google "[deligne proof of weil conjecture for k3 surfaces](https://www.google.com/search?btnG=1&pws=0&q=deligne+proof+of+weil+conjecture+for+k3+surfaces)" and click on the first (or second) link.
| 5 | https://mathoverflow.net/users/17907 | 168331 | 87,184 |
https://mathoverflow.net/questions/168263 | 4 | A ring (say unital for simplicity) is semiprimitive (or Jacobson semisimple) if its Jacobson radical is trivial, or equivalently it has faithful semisimple module. Semiprimitivity is a Morita invariant, meaning that if $R$ and $S$ are unital rings and the categories of (left) $R$-modules and $S$-modules are equivalent,... | https://mathoverflow.net/users/15934 | Is there a purely module theoretic characterization of semiprimitive rings? | I believe this answers your question, although it may not be that enlightening. Basically, you can rephrase the existence of a faithful semisimple $R$-module in category-theoretic language as follows. First, an $R$-module $M$ is faithful iff $M$ is a cogenerator in $\mbox{Mod-}R$ for the subcategory of (finitely-genera... | 3 | https://mathoverflow.net/users/11791 | 168353 | 87,191 |
https://mathoverflow.net/questions/168367 | 7 | I am looking for help pointing me in the direction of any literature or other known work that analyze the probability distribution or other important properties of random variables of the form $AB^{-1}$, where $A,B$ are independent matrix random variables whose entries from some i.i.d. random variables, such as $N(0,1)... | https://mathoverflow.net/users/7378 | Analysis of $AB^{-1}$, where $A,B$ are random matrices | A study of the eigenvalues of $AB^{-1}$ would lead you to study solutions $\lambda$. of the equation ${\rm Det}\,(A-\lambda B)=0$. This "generalized eigenvalue problem" has been studied in some detail, see for example section 6 of [How many eigenvalues of a random matrix are real?](http://www.ams.org/journals/jams/1994... | 8 | https://mathoverflow.net/users/11260 | 168369 | 87,196 |
https://mathoverflow.net/questions/168279 | 9 | Let $K/\mathbb F\_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F\_q$-automorphisms of $K$.
Obviously, $G\subseteq Aut(K)$. It is well known that
$H^1(G,K^\*) = 1$ [Hilbert 90]. But does the following hold: $H^1(Aut(K), K^\*)=1$?
Thanks in advance.
| https://mathoverflow.net/users/33128 | Variant of Hilbert 90 for Galois extensions | This is an application of Artin's Lemma to reduce to Hilbert 90. Namely, let $K$ be any field at all, and $G$ any *finite* subgroup of ${\rm{Aut}}(K)$ (as noted in this comments, such finiteness holds for the $K$ in the question with $G = {\rm{Aut}}(K)$). Then by Artin's Lemma the field $K$ is finite Galois over its su... | 11 | https://mathoverflow.net/users/43107 | 168397 | 87,210 |
https://mathoverflow.net/questions/168386 | 2 | When I was reading the paper:
Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305.
from <http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf>
I could not reproduce theorem 5.7:
Every solvable set (tilable) has a solution S such that every finite block occurring in S also ... | https://mathoverflow.net/users/40780 | An established proof in Wang Tile which I doubt | Note that the set of solutions of a set $A$ is a closed set (with respect to the product topology): every tiling that is not a solution of $A$ has an occurrence of a finite pattern $p$ not in $A$, and the cylinder set around $p$ (the set of all tilings with pattern $p$ on the same position) is an open set consisting on... | 7 | https://mathoverflow.net/users/23297 | 168404 | 87,212 |
https://mathoverflow.net/questions/168395 | 2 | In solving a physics problem, I came across a weird topological space constructed from $U(4)$, the group of $4\times4$ unitary matrices. I want to know the first two homotopy groups of it. Here is how it is defined:
Consider the set $X$ of matrices in $U(4)$
$$X=\{F\in U(4):F=N\Lambda\},$$
where $\Lambda$ is any diag... | https://mathoverflow.net/users/40893 | How to calculate the first and second homotopy groups of the following space constructed from $U(4)$ | Yes, you can compute them, and $\pi\_1(M) = \mathbb{Z}$, $\pi\_2(M) = \mathbb{Z}$.
The way I see it, there are three steps in the proof. For convenience, let me call $Y = U(2)\times U(2)$ the set of $N$s of the form above.
1. Instead of looking at $X/\sim$, let's look at $Y/\sim$ (where I denote with the same symbo... | 6 | https://mathoverflow.net/users/13119 | 168408 | 87,215 |
https://mathoverflow.net/questions/168402 | 9 | Work in the first order language of number theory, consisting of the symbols $\mathbf{0}$, $\mathbf{S}$, $\boldsymbol{+}$, and $\boldsymbol{\cdot}$, and let $Q$ denote Robinson's arithmetic.
By a *diophantine formula* we mean a formula in this language having the form $\exists y\_1 \dots \exists y\_m(f(x\_1, \dots, x... | https://mathoverflow.net/users/51345 | Essential incompleteness via diophantine formulas? | The answer is negative.
Consider the model $M=\langle\mathbb N\cup\{\infty\},0,S,+,\cdot\rangle$, where we put $S(\infty)=\infty$, $\infty+x=x+\infty=\infty$ for all $x\in M$, $\infty\cdot0=0\cdot\infty=0$, and $\infty\cdot x=x\cdot\infty=\infty$ for $x\ne0$. It is easy to check that $M\models Q$. In fact, $M$ satisf... | 14 | https://mathoverflow.net/users/12705 | 168412 | 87,217 |
https://mathoverflow.net/questions/168411 | 0 | I have a sequence $ a\_{n}$, and the set
$
A=\{ m \le x | m \in \{a\_{n}\}\_{n=0}^{\infty} \}
$
satisfying
$
\#A= x^\alpha+O(x^{\alpha-\varepsilon}).
$
Also I have
$
\#\{ m \le x : p|m , m \in \{a\_{n}\}\_{n=0}^{\infty} \}= \frac{x^\alpha}{p}+O(x^{\alpha-\varepsilon}).
$
for all p prime (of course those $p \ll... | https://mathoverflow.net/users/4486 | Smooth values of certain sequences | Let $A$ be the set of integers $n$ which are divisible by a prime number $p>n^\theta$, $\frac{1}{2}<\theta<1$. Then for a prime number $q<x^{1/3}$ we have
\begin{eqnarray\*}
\#\{n\leq x, n\in A, q|n\} & = & \underset{p\neq q}{\sum\_{p\leq x}}\underset{q|m}{\sum\_{m\leq\min(p^{(1-\theta)/\theta}, x/p)}}1 + \mathcal{O}(q... | 1 | https://mathoverflow.net/users/37555 | 168414 | 87,218 |
https://mathoverflow.net/questions/168396 | 9 | In what follows, all graphs $G$ are $K\_3$-divisible (all degrees even, number of edges a multiple of three) on $n$ vertices, where $n$ is not too small.
The famous Nash-Williams conjecture claims that $\delta(G) \ge \frac{3}{4}n$ would be sufficient for $G$ to have a $K\_3$-decomposition of its edges. (The constant ... | https://mathoverflow.net/users/45255 | Can I weaken the minimum degree hypothesis in Nash-Williams' triangle decomposition conjecture? | If you divide the vertex set into 3 parts, A, B and C, with respective sizes $an$, $bn$ and $cn$, and add all the edges except the ones connecting two vertices inside A and a vertex from A and C, then you get a graph with $\delta=bn$ and $|E|=(1-{a^2}-2ac) {n\choose 2}$. If $b<a$, you won't have a triangle-decompositio... | 8 | https://mathoverflow.net/users/955 | 168415 | 87,219 |
https://mathoverflow.net/questions/168413 | 1 | In my research I have come up with the following notion which I would like to learn more about. It may be very naive.
Let $R$ be a ring, $M$ an $R$-module and $S$ a class or $R$-modules closed under coproducts. We say that $M$ is *projective relative to $S$* if every surjection from an element of $S$ to a coproduct o... | https://mathoverflow.net/users/5181 | Modules "projective in a subcategory" | More generally, the following is true:
Let $A$ be a finite-dimensional $k$-algebra ($k$ a field), $M$ a finite-dimensional indecomposable (right) $A$-module, and $S$ the category of coproducts of copies of $M$. Then $M$ is "strongly projective" relative to $S$.
In your example, $M=k[G/H]$ is indecomposable, since, ... | 5 | https://mathoverflow.net/users/22989 | 168418 | 87,222 |
https://mathoverflow.net/questions/168381 | 9 | Several companies (google, IBM etc.) are known to hire mathematicians with advanced degrees (Ph.D. and so) to work in different sectors; I mean mostly research oriented positions in industry but could be something else too. But I almost never find any such job advertisements in the standard mathjob sites I know of, inc... | https://mathoverflow.net/users/35936 | Sources to find industrial jobs for mathematicians with a Ph.D., or with advanced backgrounds? (preferably research positions but not necessarily) | One of the best sources available now is the SIAM site:
<http://jobs.siam.org/c/search_results.cfm?vnet=0&site%5Fid=686&str=26>
This has quite a mix of jobs at various levels, but you will find research positions there as well.
| 4 | https://mathoverflow.net/users/16042 | 168425 | 87,226 |
https://mathoverflow.net/questions/168424 | 2 | Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally stratified by the (even) rank of the matrices. $Pf$ is the locus of rank at most $2n-2$, the singular locus $Sing(Pf)$ i... | https://mathoverflow.net/users/4096 | dimensions of strata of Pfaffian varieties | The codimension of forms of rank $2n-2k$ is $k(2k-1)$. In fact, it is easy to construct a resolution for each strata (a projective bundle over appropriate Grassmannian).
| 3 | https://mathoverflow.net/users/4428 | 168431 | 87,229 |
https://mathoverflow.net/questions/168382 | 3 | Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a reductive group defined over $\mathbb{F}\_p$. For any $d\in\mathbb{Z}^+$, let $C\_d(G)$ be the set of conjugacy classes of elementary abelian $p$-subgroups of $G(\mathbb{F}\_{q})$ of maximal rank, where $q=p^d$.
>
> Is $S(G)=\display... | https://mathoverflow.net/users/32261 | Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type | You might find it useful to look at some papers by Evgenii Vdovin. He has a bunch of papers discussing the maximal *orders* of nilpotent and abelian subgroups of finite groups of Lie type.
Along the way he also calculates the $p$-rank of the finite groups of Lie type. In the process (I belive) he displays an element... | 2 | https://mathoverflow.net/users/801 | 168439 | 87,233 |
https://mathoverflow.net/questions/168446 | 5 | When using axiom of choice in proofs, people often say that this is *non-constructive* because AC gives us only proofs of existence, without giving explicit example. However, because in $L$ AC holds, we have that for every existence proof using choice there is an explicit example for that in L. So AC is non-constructiv... | https://mathoverflow.net/users/30186 | Choice principle strong enough to defy $V=L$ | There are principles which imply the axiom of choice, and are inconsistent with $V=L$.
For example, the statement "*Every well-ordered chain of cardinals between $X$ and $\mathcal P(X)$ is finite*" implies the axiom of choice.1 Couple that with "*Every finite $n$ is embeddable in the cardinals between $X$ and $\mathc... | 10 | https://mathoverflow.net/users/7206 | 168450 | 87,237 |
https://mathoverflow.net/questions/168399 | 16 | Consider a degree $n$ polynomial $P(x)$ with coefficients $c\_i \in \{-1,0,1\}$ chosen uniformly and independently.
>
> What is the probability that $P(x)$ has a root which is a root of
> unity?
>
>
>
Previously asked at <https://math.stackexchange.com/questions/798082/probability-a-polynomial-has-a-root-whic... | https://mathoverflow.net/users/45564 | Probability a polynomial has a root which is a root of unity | As discussed in comments, I think for large $n$ the probability that it has a root which is a root of unity is double the probability that 1 is a root. For large $n$, $P(1)$ is a random variable whose distribution is approximately normal and whose variance is $\sigma^2=2n/3$. The probability that $P(1)=0$ is then appro... | 16 | https://mathoverflow.net/users/nan | 168457 | 87,238 |
https://mathoverflow.net/questions/168483 | 2 | Let $M$ be an odd-dimensional manifold. An *almost contact metric structure* on $M$ is a 4-tuple $(\xi, \eta, \phi, g)$, where $\xi$ is a vector field, $\eta$ a one-form, $\phi$ an endomorphism of the tangent bundle, and $g$ a Riemannian metric, all enjoying the following conditions:
\begin{aligned}
\phi^2 &= -I +\xi... | https://mathoverflow.net/users/41626 | Pseudohermitian Structures and Contact Metric Structures | Consider the (1,2) tensor Q defined by
$Q\_{jk}^i=\nabla\_k \phi\_j^i+\xi^i\phi\_j^r\nabla\_k \eta\_r+\phi\_r^i \nabla\_k \xi^r \eta\_j$
Define $J$ by restricting $\phi$ to the kernel of $\eta$. Then $ (M,\eta,J)$ is a strictly pseudoconvex, integrable CR manifold if and only if $Q=0$. This result was proved by Tan... | 0 | https://mathoverflow.net/users/48356 | 168490 | 87,246 |
https://mathoverflow.net/questions/168501 | 6 | Given a function $ g $ entire on the whole complex plane $ C $, it is possible to find an entire function $f $ such that $ f(z+1) -f(z)=g(z) $. The proof can be given using riemann surface,automorphy,covering,etc. Can anyone find a elementary proof which avoids all such things.
| https://mathoverflow.net/users/30081 | searching for an elementary proof a complex analysis result | Let $L$ be your difference operator: $(Lf)(z)=f(z+1)-f(z)$.
Consider these polynomials
$$P\_n(z)=\frac{1}{n!}z(z-1)\ldots(z-n+1),\quad n=0,1,2,\ldots.$$
Simple computation shows that $LP\_n=P\_{n-1}$. Polynomials $P\_n$ make a basis in the space
of all polynmials, because there is one polynomial of each degree.
This a... | 7 | https://mathoverflow.net/users/25510 | 168502 | 87,250 |
https://mathoverflow.net/questions/168433 | 7 | Given a spectrum, is there any kind of machinery that can tell you whether it is the K-theory spectrum of some recognizable category? For example, could TMF be realized in this way? In this case we would expect the chromatic height to decrease by one by the chromatic redshift conjecture.
| https://mathoverflow.net/users/27526 | Is this spectrum the algebraic K-theory spectrum of something? | The standard algebraic $K$-theory constructions give connective spectra. So $TMF$ is not going to be in the image of such a construction. However Thomason showed that every connective spectrum can be constructed as the algebraic $K$-theory of some symmetric monoidal category (See Mandell's recent paper for a new proof ... | 15 | https://mathoverflow.net/users/8818 | 168504 | 87,251 |
https://mathoverflow.net/questions/168513 | 6 | Does there exist an [Einstein manifold](http://en.wikipedia.org/wiki/Einstein_manifold) which is not conformally flat, which is to say one which has non-vanishing [Weyl tensor.](http://en.wikipedia.org/wiki/Weyl_tensor) If so, what is a good example.
| https://mathoverflow.net/users/51325 | Example of Non-Conformally Flat Einstein Manifold? | All conformally flat homogeneous riemannian manifolds are symmetric spaces, by a result of Takagi. All homogeneous riemannian manifolds of dimension $\leq 11$ admit Einstein metrics, by results of Wang and Ziller. Most homogensous riemannian manifolds are not symmetric spaces.
References:
-----------
Takagi, H.: Co... | 5 | https://mathoverflow.net/users/394 | 168517 | 87,253 |
https://mathoverflow.net/questions/168520 | 9 | When writing with a [friend](http://googology.wikia.com/wiki/User%3aWythagoras) of mine today we came up with idea of extending ITTM concept of Hamkins and Kidder. First of all, I am familiar with one of Hamkins and Lewis results saying that every machine either halts or repeats before $\omega\_1$, so allowing standard... | https://mathoverflow.net/users/30186 | Uncountable time Turing machines | I introduced a very similar model in my paper [Cardinal-Recognizing Infinite Time Turing Machines](http://arxiv.org/abs/1310.5590), where I add a state which fires at cardinal times. It is easy to see that your model and mine are computationally equivalent, although the precise values of the ordinals $\gamma,\lambda,\S... | 7 | https://mathoverflow.net/users/1058 | 168524 | 87,256 |
https://mathoverflow.net/questions/168511 | 1 | Let $(M,\omega)$ be a Kähler manifold with a [pre-quantum Line bundle](http://ncatlab.org/nlab/show/prequantum+line+bundle) $L$ and
$\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is for the flag variety $G/P$ when $\frac{\text{Aut}(G/P,L)}{S^1}$ i... | https://mathoverflow.net/users/nan | When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete? | Without knowing anything in particular about the $S^1$-action, your condition seems to me unlikely to be satisfied very often. Let $G=SL\_2(\mathbb{C})$ and let $P$ be the standard Borel of upper-triangular matrices. Given any weight $n\in\mathbb{Z}$ of the standard maximal torus $T$, we can construct an associated lin... | 0 | https://mathoverflow.net/users/25358 | 168527 | 87,258 |
https://mathoverflow.net/questions/168529 | 6 | Let $p\_1=2, p\_2 = 3,\ldots,$ be the prime numbers, and define $n\_i = \prod\_{j=1}^i p\_j$. Moreover, let $E\_i $ be the elliptic curve defined by $y^2 = x^3 + n\_i$.
Can one compute the torsion group $E\_i(\mathbb Q)\_{tors}$ in terms of $i$? Or at least its cardinal?
I expect it to be trivial. I checked this f... | https://mathoverflow.net/users/51401 | Torsion group of the following elliptic curve | Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves gives a classification of the torsion subgroup of $E : y^{2} = x^{3} + D$ if $D$ is a sixth-power free integer. The torsion subgroup is trivial unless $D$ is a square, a cube, or $-432$.
One way to prove this is to show that if $p \equiv 2 \pmod... | 8 | https://mathoverflow.net/users/48142 | 168532 | 87,261 |
https://mathoverflow.net/questions/168488 | 0 | Let $\{c\_n\}$ be a descending sequence of positive real numbers, and let $\{X\_i\}$ be a sequence of i.i.d. random variables.
Are the following statements equivalent?
1. $\operatorname{E}(X\_1^2) < \infty$ and $\operatorname{E}(X\_1) = 0$ implies $\lim \limits\_{n \to \infty} c\_n \sum \limits\_{i = 1}^n X\_i \xri... | https://mathoverflow.net/users/51189 | strong law of large number | The [Law of the Iterated Logarithm](http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm) tells you
$$
\limsup\_{n\to\infty}\left|\frac{S\_n}{\sqrt {n\log\log n}}\right|=\sigma\sqrt 2\text{ with probability 1,}
$$
where $S\_n:=X\_1+\ldots+X\_n$, and $\sigma^2=\mathbb EX\_1^2$. Hence if
$\sqrt{n\log\log n}\cdot ... | 4 | https://mathoverflow.net/users/11054 | 168533 | 87,262 |
https://mathoverflow.net/questions/168422 | 1 | Somebody tell me that:
For a bundle(maybe polystable) over algebraic manifold, take a symmetric power of the bundle and tensor with its determinant line bundle to some power.
Assume that the resulting bundle has zero first chern class.Then if it has sections, the structure group reduces to smaller group.
Who can t... | https://mathoverflow.net/users/40042 | characterization of structure group | The correct version of this statement is the following. Let $B$ be a stable vector bundle with structure group $G$, and $TB$ a tensor component in the bundle of all tensors over $B$ (that is, a sub-bundle determined by a symmetric group representation) with trivial determinant. Then holomorphic sections of $TB$ are $G$... | 2 | https://mathoverflow.net/users/3377 | 168538 | 87,264 |
https://mathoverflow.net/questions/168478 | 16 | While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:
**Question:** Do there exist a non-trivial decomposition $\mathbb{R} = T \oplus T'$ as an additive $\mathbb{Q}$-vector space and another decomposition $\mathbb{R}^{+}... | https://mathoverflow.net/users/7894 | A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space | Yes. The field of formal Puiseux series over $\overline{\mathbb Q}$ is an algebraically closed field of characteristic $0$ with the cardinality of the continuum, hence is isomorphic to $\mathbb C$. It is easy to check that the ring of Puiseux series with nonnegative valuation does not contain $\mathbb R$, hence its int... | 18 | https://mathoverflow.net/users/18060 | 168541 | 87,266 |
https://mathoverflow.net/questions/168167 | 4 | I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw [this post](https://mathoverflow.net/questions/78106/solving-a-quadratic-matrix-equation), but unfortunately it seems that no solution was y... | https://mathoverflow.net/users/51260 | Solving a quadratic matrix equation with fat matrix | I assume that the matrices are real, we know $ X $ and we seek a solution in $ T $.
Case 1.$ rank(X)>n $.There are no solutions in $ T $.
Case 2.$ rank(X)\leq n $.There is an orthogonal matrix $ P$
and a diagonal matrix $ D=diag(\lambda\_1,\cdots,\lambda\_n,0,\cdots,0) $ s.t. $ X=PDP^T $ and
$\lambda\_i\geq 0 $ ... | 1 | https://mathoverflow.net/users/51189 | 168545 | 87,269 |
https://mathoverflow.net/questions/165834 | 5 | Let $n$ be a natural number.
Let $U \subseteq \mathbb{F}\_2^n$ be a linear subspace of dimension $k$.
What is the maximum number of vectors in $U$ of Hamming weight $\ell$?
The case I am specifically interested in is $k \approx 0.99n$, $\ell = n/2$, where $n$ is a large enough even number. Is this number at most $C \... | https://mathoverflow.net/users/9017 | Number of binary vectors of a given Hamming weight in a subspace of the Hypercube | I just found out that the question (at least with the set of parameters written above) was asked by Ben-Or and answered by Linial and Samordinsky.
Their paper proves that for any constant $r>1/2$ a subspace of dimension $k = rn$ has at most $C\_r \cdot 2^k \cdot \frac{\binom{n}{n/2}}{2^n}$ vectors of weight $n/2$.
<htt... | 1 | https://mathoverflow.net/users/9017 | 168547 | 87,270 |
https://mathoverflow.net/questions/168489 | 1 | I asked this question on math stack exchange, but I didn't get any responses. So, now I am motivated to ask it here. Is the class of free monoids first order axiomatizable? And what about the class of full transformation monoids, in other words monoids that are isomorphic to a monoids of all functions over a set S to i... | https://mathoverflow.net/users/43439 | Free monoids and full transformation monoids | In order to prove that something is not first-order axiomatizable it is sometimes fun to use ultraproducts. I think we can do this in this case. Note that one of the characteristic features of free monoids is that any element of a free monoid can have only finitely many "prefixes" (this is number 4 from Ben Steinberg's... | 4 | https://mathoverflow.net/users/6485 | 168554 | 87,272 |
https://mathoverflow.net/questions/168549 | 1 | I am not sure whether the title is appropriate for this question or not. I am sorry if there is anyone who is confused with the title and the contents.
What I want to ask is the following: let $k$ be a field and $\bar{k}$ be a separable closure of $k$. Let $X$ be a 'good' scheme over $k$ (you can give any condition o... | https://mathoverflow.net/users/44006 | Descend of etale morphism | That is often impossible. I am pretty certain all of the following has been explained here before, but here it is again.
Begin with a smooth, projective $k$-curve $P$ such that $\overline{P}$ is isomorphic to $\mathbb{P}^1\_{\overline{k}}=\text{Proj}\ \overline{k}[s,t]$, yet $P$ is not isomorphic to $\mathbb{P}^1\_k$... | 1 | https://mathoverflow.net/users/13265 | 168555 | 87,273 |
https://mathoverflow.net/questions/168552 | 2 | Consider two functions $\alpha,\beta: \mathbb{R}^2 \to \mathbb{R}$, where $\alpha$ is given and we look for solutions $\beta$ such that
$$\*(d\alpha \wedge d\beta) = \lambda \beta$$
for some $\lambda \in \mathbb{R}$ and where $d$ is the exterior derivative, $\*$ is the Hodge star operator, which turns the 2-form $d... | https://mathoverflow.net/users/43839 | Eigenvalue problem of an operator involving the exterior derivative of differential forms | No, the problem as stated is not well posed. You need initial data.
We can rewrite the above equation in coordinates to get $$ \partial\_x \alpha \partial\_y \beta - \partial\_y \alpha \partial\_x \beta = \lambda \beta $$ This is a first order PDE and so one can use method of characteristics to solve this. I know thi... | 2 | https://mathoverflow.net/users/49404 | 168560 | 87,275 |
https://mathoverflow.net/questions/168528 | 2 | Let $(A,m,k)$ be commutative noetherian local ring.
Can one detect if $A$ is a Gorenstein ring from the [André–Quillen homologies](http://en.wikipedia.org/wiki/Andr%C3%A9%E2%80%93Quillen_cohomology) $H\_n(A,k,-)$?
| https://mathoverflow.net/users/51400 | Can André–Quillen homology detect the property of being Gorenstein? | "No" if you want a standard vanishing result, and "no up to now" if you are thinking in another kind of characterization (see however for a related result: Garcia-Soto, Ascent and descent of Gorenstein property, Glasgow Math. J.
46 (2004) 205–210).
| 1 | https://mathoverflow.net/users/36672 | 168570 | 87,282 |
https://mathoverflow.net/questions/168562 | 3 | I am reading the paper of [Convergence of the Yamabe flow for arbitrary initial energy](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-69/issue-2/Convergence-of-the-Yamabe-flow-for-arbitrary-initial-energy/10.4310/jdg/1121449107.full)
I am stuck by one part of the paper. Suppose $u\_\infty... | https://mathoverflow.net/users/22815 | Implicit function theorem for operator | Consider the map $F:C^{2,\alpha}\to \mathbb{R}^A \times (C^{0,\alpha}\cap Im \Pi)$
$$
F: f \mapsto \left( \left(\int u\_\infty^{\frac{4}{n-2}}(f-u\_\infty)\psi\_a\right)\_{a\in A},\Pi\left( \frac{4(n-1)}{n-2} \Delta f - Rf +r f^\frac{n+2}{n-2}\right)\right).
$$
Compute $DF(u\_\infty)$:
$$
DF(u\_\infty)f = \left( \left(... | 2 | https://mathoverflow.net/users/1540 | 168574 | 87,283 |
https://mathoverflow.net/questions/168373 | 3 | I am currently struggeling with the following differential equation:
$$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$
where $a \in \mathbb{R}$ constant, $\phi \in \mathbb{N}$ is a parameter and $\lambda$ is the eigenvalue.
Now, I noticed the following:
for $\phi = 1$ we ha... | https://mathoverflow.net/users/nan | What does this ODE have to do with the associated Legendre polynomials? | According to Maple, the general solution is in terms of HeunC functions:
$$ f \left( t \right) ={\it \\_C1}\,{\it HeunC} \left( 4\,a,-1/2,-1/2,-2\,
a\phi,3/8-{a}^{2}+a\phi-\lambda,{t}^{2} \right) +{\it \\_C2}\,{\it
HeunC} \left( 4\,a,1/2,-1/2,-2\,a\phi,3/8-{a}^{2}+a\phi-\lambda,{t}^{2
} \right) t
$$
For $\phi = 3$ (wi... | 3 | https://mathoverflow.net/users/13650 | 168590 | 87,287 |
https://mathoverflow.net/questions/168521 | 3 | in Grothendieck's Tohoku paper (page 127), Grothendieck asserts that the category of holomorphic vector bundles $\mathbf{Bund}(X)$ over a fixed Riemann surface $X$ does not satisfy $\mathbf{AB2}$, but satisfies $\mathbf{AB1}$. Explicitly, he claims that $\mathbf{Bund}(X)$ has kernels and cokernels, however the canonica... | https://mathoverflow.net/users/40883 | Holomorphic vector bundles over a Riemann surface does not satisfy $\mathbf{AB2}$ but satisfies $\mathbf{AB1}$ | (This should really be a comment on S. Carnahan's answer, but don't have enough reputation.)
All coherent sheaves on a Riemann surface split into the direct sum of a torsion sheaf and a vector bundle. This means we get a pair of adjoint functors, the forgetful functor $f:Bund(X)\rightarrow Coh(X)$ and the "kill-torsi... | 5 | https://mathoverflow.net/users/51424 | 168592 | 87,289 |
https://mathoverflow.net/questions/168593 | 6 | Numerical evidence for the first hundred Apery numbers
$$A\_n=\sum\limits\_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2$$
suggests the following congruence relation
$$A\_n\equiv 0\; (\mathrm{mod}\; 5),\;\;\mathrm{if}\;\; n\equiv \{1,3\}\;(\mathrm{mod}\; 5).$$
Was this congruence ever proved?
| https://mathoverflow.net/users/32389 | Conjectured congruence for the Apery numbers | The Apery numbers satisfy the recurrence
$$
n^3 A\_n = (34n^3-51n^2+27n-5)A\_{n-1}- (n-1)^3 A\_{n-2}.
$$
If $n\equiv 1\pmod 5$ this recurrence gives
$$
A\_n \equiv (4\cdot 1-1\cdot 1+2\cdot 1-5)A\_{n-1} \equiv 0 \pmod 5.
$$
If $n\equiv 3 \pmod 5$ it gives
$$
2 A\_n \equiv (4 \cdot 2-1\cdot 4+2\cdot 3 -5)A\_{n-1... | 5 | https://mathoverflow.net/users/38624 | 168594 | 87,290 |
https://mathoverflow.net/questions/168596 | 8 | I asked this [question](https://math.stackexchange.com/q/813334/660) on Mathematics Stack Exchange, but got no answer:
>
> Given two vector spaces $V$ and $W$ over a field $K$, what is the dimension of $\operatorname{Hom}\_K(V,W)\ $?
>
>
>
To state the partial result I've been able to obtain, let me introduce ... | https://mathoverflow.net/users/461 | dim Hom(V,W) =? | Using $\hom\_K(V,W)=W^\alpha$ and
[Dimension of infinite product of vector spaces](https://mathoverflow.net/questions/49551/dimension-of-infinite-product-of-vector-spaces)
we get $\dim \hom\_K(V,W)=(\#W)^\alpha=((\# K)\cdot\beta)^\alpha$.
| 9 | https://mathoverflow.net/users/12166 | 168599 | 87,292 |
https://mathoverflow.net/questions/168519 | 2 | Consider the problem of finding $u \in L^2(0,T;H^1)$ with $u' \in L^2(0,T;L^2)$ such that
$$\int\_0^T \int\_{\Omega}u'(t)\varphi(t) + \int\_0^T \int\_{\Omega}\nabla (F(u(t)))\nabla \varphi(t) = \int\_0^T \int\_\Omega f(t)\varphi(t)$$
where $F$ is differentiable and $F'$ bounded above and below away from $0$, and $F(0)=... | https://mathoverflow.net/users/51399 | Existence of solution with $u' \in L^2(0,T;L^2)$ to a nonlinear parabolic PDE | Given your setting, you cannot hope to get the strong regularity $\partial\_t u\in L^2(0,T;L^2)$ for free. Indeed you're solving the generalized Porous Media Equation
$$
\partial\_tu=\Delta F(u)
$$
with zero Dirichlet boundary conditions. The fundamental reason is that $F(u)\in L^2(0,T;H^1\_0)$ is really the energy spa... | 2 | https://mathoverflow.net/users/33741 | 168603 | 87,294 |
https://mathoverflow.net/questions/168128 | 12 | In an article I'm writing I want to quote (with attribution) the original version of an aphorism that says that one can often find mathematical gold in the waste-baskets of physicists. Would someone provide a reference to that original version?
Note added after awarding bounty: For the time being, I'm going to assume... | https://mathoverflow.net/users/3621 | Source of quotation about the waste-baskets of physicists | Here's one scientist (not quite a mathematician) who found gold in wastebaskets:
*I started looking in the trash cans of science for such phenomena [fractal scaling], because I suspected that what I was observing was not an exception but perhaps very widespread.*
[Benoit Mandelbrot](http://faculty.cbu.ca/philosoph... | 7 | https://mathoverflow.net/users/11260 | 168616 | 87,300 |
https://mathoverflow.net/questions/168585 | 5 | Euler Characteristic of Sheaves and the Generalized Gauss-Bonnet Theorem
========================================================================
Consider a sheaf $\mathscr{F}$ over a site $\mathscr{(C,J)}$, where $\mathscr{C}$ is a small category with a coverage $\mathscr{J}$. The Euler characteristic is defined as ... | https://mathoverflow.net/users/nan | The Gauss-Bonnet theorem for Sheaves | There is an answer in the case of constructible sheaves. The whole story can be found in Kashiwara and Shapira's book "Sheaves on Manifolds" Chapters VIII and IX.
I will try to summarize this rather long story. First, a prototype for a constructible sheaf is a locally constant sheaf supported on a *reasonable set*, e... | 9 | https://mathoverflow.net/users/20302 | 168617 | 87,301 |
https://mathoverflow.net/questions/168613 | 4 |
>
> *Is it true that a sequence of real numbers $\{a\_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a\_n\}$ $(h \in \mathbb{R})$ all converge to zero?*
>
>
>
In case the answer is affirmative (and please skip the obvious implication):
**1.** How far can we replace the condition "for every va... | https://mathoverflow.net/users/21123 | A question on null sequences | The following theorem about uniform distribution on the circle $\mathbb T^1$ is well known (and is due to Weyl): given an increasing sequence of positive integers $k\_n$, there is $z\in\mathbb T^1$ such that $z^{k\_n}$ is uniformly distributed in $\mathbb T^1$ (using the complex notation). In fact, this statement is tr... | 8 | https://mathoverflow.net/users/50457 | 168618 | 87,302 |
https://mathoverflow.net/questions/168625 | 4 | I am reading Ravenel's *Localization with Respect to Certain Periodic Homology Theories* where he states;
>
> For $n\ge2$, the spectra E(n)
> represent periodic homology theories which at present have no known geometric interpretation comparable to the description of K-theory in terms of
> vector bundles.
>
> ... | https://mathoverflow.net/users/27526 | Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory? | The answer is no, at least as far as I am aware of. For height 2 case, you can
consider elliptic cohomology as some sort of geometric interpretation of $E(2)$
theory, but this is far less straight-forward than the vector bundle description
of K-theory. Furthermore, the discovery by Igor Kriz of a group G with
$K(n)^{od... | 5 | https://mathoverflow.net/users/43326 | 168629 | 87,307 |
https://mathoverflow.net/questions/168280 | 10 | A [W-type](http://ncatlab.org/nlab/show/W-type) is an [initial algebra](http://ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor) for a [polynomial endofunctor](http://ncatlab.org/nlab/show/polynomial+functor) $P$ on a category $C$. A well-known example is that of a [natural numbers object](http://ncatlab.org/nla... | https://mathoverflow.net/users/4177 | What sort of W-types follow from existence of an NNO? | If CC is infinitary lextensive and in addition locally small, then the global sections functor $\Gamma$ : CC --> Set has a lex left adjoint $\Delta$ : Set --> CC, and CC automatically has a natural numbers object since NNOs are preserved by inverse image functors. Moreover, the reason that this is the case can be formu... | 6 | https://mathoverflow.net/users/51432 | 168634 | 87,312 |
https://mathoverflow.net/questions/164092 | 43 | We see that $$\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\prod\_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\prod\_p \left(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)}\right)=\prod\_p\left(\frac{p^2-1}{p^2+1}\right)$$
$$\implies \prod\_p \left(\frac{p^2-1}{p^2+1}\right)=\frac{2}{5},$$
But is this t... | https://mathoverflow.net/users/47897 | Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function? | I found a proof of
$$5 \sum\_{m=1}^{\infty} \frac{1}{m^4} = 2 \left( \sum\_{n=1}^{\infty} \frac{1}{n^2} \right)^2$$
by rearranging sums and [wrote it up](http://www.math.lsa.umich.edu/~speyer/ZetaFun2.pdf). The argument is just 1.5 pages, the other 4.5 are explanations and context.
Here is a summary using divergent ... | 60 | https://mathoverflow.net/users/297 | 168650 | 87,315 |
https://mathoverflow.net/questions/168641 | 4 | Let $g(n,k)$ be the number of chains
$$ A\_k \subset A\_{k-1} \subset\dots\subset A\_1 \subset A\_0 $$
of $k$ *proper* subset inclusions, where $A\_k\neq\emptyset$ and $A\_0$ is a standard $n$-element set. Then
$$ \sum\_{k\ge 0} (-1)^k g(n,k) = (-1)^{n-1}. $$
I can prove this by a fairly boring and unilluminat... | https://mathoverflow.net/users/49 | Counting chains of inclusions | A chain
$$ A\_k \subset A\_{k-1} \subset\dots\subset A\_1 \subset A\_0 $$
can be represented by the ordered partition $(B\_1, B\_2, \dots, B\_{k+1})$ of the set $A\_0=\{1, 2, \dots, n\}$ where $B\_1=A\_k$, $B\_2=A\_{k-1}-A\_k$, $\dots,$ $B\_{k+1}=A\_0-A\_1$.
First, a generating function proof. If we wanted to count ... | 7 | https://mathoverflow.net/users/10744 | 168651 | 87,316 |
https://mathoverflow.net/questions/168648 | 2 | Suppose that $G$ is a group and that $H$ is a subgroup, both finitely generated, and assume that there is a non-trivial H-almost invariant set $X$ with $HXH=X$. Kropholler's Conjecture asserts that $ G $ splits over a subgroup commensurable with a subgroup of $H$.
Note that the algebraic hypothesis $HXH= X $ can be r... | https://mathoverflow.net/users/51189 | Kropholler's Conjecture and 3-manifolds | The paper of Scott and Swarup mentioned in the question cites Kropholler and Roller's 'Splittings of Poincaré duality groups'. On page 35 they write:
>
> 'Let $K$ be a Poincaré duality group of dimension $(n−1)$ which is a subgroup of a Poincaré duality group $G$ of dimension $n$ ... Kropholler and Roller defined a... | 5 | https://mathoverflow.net/users/1463 | 168652 | 87,317 |
https://mathoverflow.net/questions/168644 | 0 | Let $$L(C,s)=\sum\_{n=1}^\infty \frac{a\_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-function of a holomorphic cusp form for a congruence subgroup and it is entire function and have a holomorphic continuation.... | https://mathoverflow.net/users/25947 | Is there is a known relation or expression containing the algebraic rank $r$? | First, is the reference supposed to be to the book "Rational Points on Elliptic Curves" by John Tate and myself? The formula
$$2^r=\frac{\#\alpha(\Gamma)\cdot\#\overline\alpha(\Gamma)}{4}$$
appears at the bottom of page 91 of our book. So maybe Dr. Gica's notes are based on that?
Anyway, this is all a special case of... | 7 | https://mathoverflow.net/users/11926 | 168655 | 87,319 |
https://mathoverflow.net/questions/168438 | 7 | I am looking for references (with proof) for the following statement:
>
> *Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p\_t(x , y)$ be the heat kernel of $M$. Let $K$ be compact in $M \times M$ such that $K$ is disjoint
> from the cut locus and let $T>0$. Then there exist constants $c, C ... | https://mathoverflow.net/users/16702 | Sharp Gaussian upper bounds on Heat Kernel | If $\mathbb{M}$ is a compact manifold, the bounds you are interested in are a consequence of the Minakshisundaram-Pleijel expansion
$p\_t(x,y)\sim\_{t \to 0} \frac{e^{-\frac{d^2(x,y)}{4t}}}{(4\pi t)^{n/2}}\sum\_{k=0}^{+\infty} u\_k(x,y)t^k $
which holds uniformly on compact sets of $\mathbb{M}\times \mathbb{M} -C\_... | 6 | https://mathoverflow.net/users/48356 | 168657 | 87,320 |
https://mathoverflow.net/questions/168649 | -1 | Let $K$ be a field of characteristic zero and $X$ be a projective variety over $\mbox{Spec} K$. Denote by $\bar{K}$ the algebraic closure of $K$. Let $\bar{X}$ be the fiber product $X \times\_K \bar{K}$. Suppose that $\bar{X}$ is a smooth Fano variety. Is it true that $X$ is Fano?
| https://mathoverflow.net/users/46578 | Is being fano preserved under flat base change | Let $f:X\rightarrow Y$ be a morphism of schemes over $S$. If $g :T\rightarrow S$ is faithfully flat and quasi-compact, and $F:X^{'}\rightarrow Y^{'}$ is the base changes by $g$, you have the follwing: an invertible sheaf $\mathcal{L}$ is $f$-ample if and only if its pull-back $L^{'}$ is ample.
In you case this yield... | 7 | https://mathoverflow.net/users/14514 | 168661 | 87,321 |
https://mathoverflow.net/questions/168656 | 2 | $E$ is a vector space of dimension $n \geq 2$. $\mathbb{F}=(F\_1,F\_2,\dots,F\_n)$ and $\mathbb{G}=(G\_1,G\_2,\dots,G\_n)$ are two complete flags of $E$.
We say that $(\mathbb{F},\mathbb{G})$ is in position $\sigma \in \mathfrak{S}\_n$ if and only if there exists a basis $(e\_1,\dots,e\_n)$ of $E$ such that for each ... | https://mathoverflow.net/users/41060 | Position of complete flags | Another way to characterize relative position is by looking at intersections: it is equivalent asking that $\dim(F\_i\cap G\_j)=\#\big([1,i]\cap\sigma([1,j])\big)$. How does this number change when we replace $\sigma$ by $\tau\_k\sigma$? The RHS stays the same unless $i=k$, and in that case it drops if $k\in \sigma([1,... | 4 | https://mathoverflow.net/users/66 | 168662 | 87,322 |
https://mathoverflow.net/questions/168642 | 12 | For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$
A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) as a product of disjoint cycles. To set an example we'll use $n^~\_0=8$ and
$$\sigma\_0^~=(1247)(358)(6).$$
I'm look... | https://mathoverflow.net/users/2811 | Categorical interpretation of disjoint cycle notation for tracing permutations | I'm not completely clear on what answer would be considered satisfactory, but here's one categorical way to think about tracing in $\mathbf{Bij}$.
The morphisms of the symmetric monoidal category with duals obtained by applying the Joyal-Street-Verity construction to $\mathbf{Bij}$ can be pictured in terms of string... | 7 | https://mathoverflow.net/users/2926 | 168665 | 87,324 |
https://mathoverflow.net/questions/168666 | 2 | Can one find a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every finite supersolvable group $G$ we have: $d(G') \leq f(d(G))$?
Here $d(K)$ is the cardinality of a minimal set of generators for $K$, and $K'$ is the commutator subgroup of $K$.
| https://mathoverflow.net/users/38889 | Number of generators of the commutator | By a method similar to Jeremy Rouse's comment on my original (incorrect) answer, I think the answer is no. A $p$-group is certainly supersolvable. If we take $G = C\_{p} \wr C\_{p^{n}},$ then $d(G) =2,$ $d(G^{\prime}) = p^{n}-1$ - in fact, the case $n = 1$ already seems to provide a counterexample to the question as as... | 4 | https://mathoverflow.net/users/14450 | 168667 | 87,325 |
https://mathoverflow.net/questions/168668 | 1 | I want to solve this system of N non-linear equations without using a numerical method:
$x\_{k}^{2}= \alpha\_{k }+ \sum\limits\_{m=1}^{N} (\beta\_{km} x\_{m} + \psi\_{km} x\_{m}^{\*})$
With
$\left| x\_{k}-1 \right| \leq 1 $
All values are complex and $x^{\*} $ is the conjugate of x. Any idea? An approximated so... | https://mathoverflow.net/users/51455 | System of quadratic complex equations | Put $X=[x\_1,\cdots,x\_N]^T$. We consider an equation in the form $X\circ X+A'+B'X+C'\overline{X}=0$ where $\circ$ is the Hadamard product, $A'$ is a vector and $B',C'$ are square matrices.
Step 1. Put, for every $k$, $x\_k=1+y\_k$. Then $Y$ satisfies an equation in the previous form. Then we may assume that $|x\_k|\... | 1 | https://mathoverflow.net/users/9091 | 168685 | 87,330 |
https://mathoverflow.net/questions/168672 | 6 | Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}\_q$.
The Grothendieck group of the category of finitely-generated $\mathbf{F}\_q[G]$-modules is naturally identified with a group of $\mathbf{C}$-valued functions on the set of co... | https://mathoverflow.net/users/51456 | What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules? | If I remember correctly, it is isomorphic to the group of ordinary (virtual) characters over $K=\mbox{Frac}(\mathcal{O})$, the field of fractions of $\mathcal{O}$. That is, the Grothendieck group of $KG$. An isomorphism is given by simply tensoring with $K$, so that $\mathcal{O}G$-modules $M,N$ are identified in $K\_0(... | 5 | https://mathoverflow.net/users/26223 | 168692 | 87,333 |
https://mathoverflow.net/questions/168684 | 2 | Given the divergent integral
$$ \int \_{0}^{\infty}dx \int\_{0}^{\infty}dy \frac{x^{2}y+1}{1+x+y} $$
how can I apply Hadamard's finite part to give a finite meaning to it ?
It is just made by applying the itetrated finite part first to '$dx$' and then '$dy$'?
| https://mathoverflow.net/users/23964 | double integral and Hadamard finite part | The Hadamard finite part description is strictly defined only for $1$-dimensional integrals. Also, it is usually defined for singularities within the domain of integration or at a finite boundary. On the other hand, your integrals is divergent at infinity. It is of course possible to try to do something similar in this... | 2 | https://mathoverflow.net/users/2622 | 168697 | 87,334 |
https://mathoverflow.net/questions/168687 | 1 | Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm.
The Laplace-Belrami-Operator $\Delta$ on $X$ with domain $C^2(M)$ is not densely defined, as $C^2$ is not dense in $\mathcal{M}(X)$.
**Question:** Can the Laplacian be ... | https://mathoverflow.net/users/16702 | Laplacian on space of measures | The semigroup exists --- it is the adjoint of the heat semigroup on $C(X)$. However, it is *not* strongly continuous at $t=0$ and thus does not have a densely defined generator. The semigroup for $t>0$ is given by
$$e^{t\Delta}\mu(E) \ = \ \int\_{E} H\_t(x,y) d V(x) d\mu(y) , $$
where $H\_t(x,y)$ is the heat kernel on... | 4 | https://mathoverflow.net/users/6781 | 168707 | 87,336 |
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