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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/64211 | 16 | Roth's theorem on 3-term arithmetic progressions (3AP) is concerned with the value of $r\_3(N)$, which is defined as the cardinality of the largest subset of the integers between 1 and N with no non-trivial 3AP. The best results as far as I know are that
$CN(\log\log N)^5/\log N \ge r\_3(N) \ge N\exp(-D\sqrt{\log N})... | https://mathoverflow.net/users/14934 | Roth's theorem and Behrend's lower bound | Dear Yui,
It's only slightly more than a casual remark. Our inability to find a better example is certainly a big reason for believing that Behrend's bound is correct. Julia Wolf and I slightly rehashed the proof of Behrend's bound
<http://arxiv.org/abs/0810.0732>
When formulated this way, I think the constructio... | 19 | https://mathoverflow.net/users/5575 | 64218 | 39,676 |
https://mathoverflow.net/questions/64158 | 1 | Consider the following second order linear ODE with mixed boundary condition:
$$\frac{d^2f}{dt^2}+a(t)f(t)=0,~\frac{df}{dt}(0)=u,~f(1)=0,$$
where $u\in R$ and $a\in C[0,1]$ are fixed.
Is the solution to this equation unique? If so, how to prove it? Thanks!
| https://mathoverflow.net/users/14462 | Second order linear ODE with mixed boundary condition | I'm not sure whether this is really appropriate for MathOverflow or not. Still, let me say a little more: As I've mentioned above, you can work out completely the case where $a$ is constant using explicit solutions. If $a$ is not constant, I'm not aware of a definitive answer but you can get separate necessary conditio... | 1 | https://mathoverflow.net/users/613 | 64222 | 39,679 |
https://mathoverflow.net/questions/64205 | 6 | I have a question regarding ergodicity in infinite dimensional spaces.
Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by the Bochner-Minlos theorem.
We can define the translation $\tau\_x \phi$ of a distribution $\phi$ by:
$$ \langle \tau\_... | https://mathoverflow.net/users/4047 | Ergodicity of Convoluted White Noise | The answer is yes for both situations because mixing implies ergodic.
To make things precise, let $S=S(\mathbb{R}^n)$ be the Schwartz space of
smooth real valued functions on $\mathbb{R}^n$ which decay faster than any negative power of the
Euclidean norm. Let $S'$ be the dual space of tempered distributions.
For $f\in ... | 7 | https://mathoverflow.net/users/7410 | 64224 | 39,681 |
https://mathoverflow.net/questions/51654 | 12 | One of the reasons I think ultrapowers are interesting is the following corollary of Łoś's theorem:
>
> Let $V$ be a relational structure and $^\*V$ an ultrapower of $V$. Then a first order statement about $V$ is true if and only if the corresponding statement for $^\*V$ is true, where the relations $R$ on $V$ are ... | https://mathoverflow.net/users/6481 | ultrapowers and higher order logic | Look. Suppose you take an infinite sequence of sets $ A\_0, A\_1, A\_2, \dots $ (with no structure apart from equality, that is, no relations or functions), such that $ A\_n $ has $ n $ elements. If you take a ultraproduct of these sets (using a non-principal ultrafilter over their indices), can the product be a finite... | 8 | https://mathoverflow.net/users/5340 | 64235 | 39,689 |
https://mathoverflow.net/questions/64237 | 12 | Let $C$ have all limits of diagrams indexed by $J$, and for each $b\in B$ let $E\_b:C^B \rightarrow C$ the evaluation functor that evaluates at $b$.
Given a diagram $F: J \rightarrow C^B$,
we can calculate its limit by doing it pointwise, i.e. since for each $b\in B$,
$E\_b \circ F : J \rightarrow C$ has a limit $v\_... | https://mathoverflow.net/users/14800 | Limits in functor categories | What you're asking is whether every limit in a functor category $[B,C]$ is a pointwise limit. The answer is yes if C is complete, but not always otherwise. Kelly gives an example in *Basic Concepts of Enriched Category Theory*, section 3.3, of a limit in a functor category that is not a pointwise limit.
I don't know ... | 15 | https://mathoverflow.net/users/4262 | 64240 | 39,692 |
https://mathoverflow.net/questions/64233 | 2 |
```
Given a undirected and unweighted graph G(V,E). M is a subset of vertices of V.
s is a vertex in V - M.
Find an optimal tree T of G defined as:
(1) M and s are in V(T)
(2) Distance (which is length of the shortest path) from s to any vertex in M in tree T is equal to distance from s to these vertices in G
(... | https://mathoverflow.net/users/14939 | Could this be a NP complete? | The problem is NP-complete.
I think that the following algorithm describes a polynomial reduction of SAT to your problem.
Let S be an instance of SAT. So you have a finite set of clauses $C\_1$, $C\_2$, ...,$C\_n$.
and a finite set of variables $p\_1$, $p\_2$, ..., $p\_k$. Each clause contains some literals, i... | 4 | https://mathoverflow.net/users/14915 | 64241 | 39,693 |
https://mathoverflow.net/questions/64242 | 10 | This may seem a vague question, but in the process of explaining the hierarchy of transfinite sets to my students, I've had to use the Cardinality(powerset(S)) > Cardinality(S) argument, which appeared to me to be extremely weak in indicating how much larger the powerset is. Is there a better argument?
In contrast, C... | https://mathoverflow.net/users/14940 | How much larger is the powerset of a transfinite set? | For infinite sets, $A$ having larger cardinality than $B$ already implies that $A$ is "much larger," and perhaps a good way to see this is to think of all the ways of making $B$ "larger" which *don't* increase its cardinality. E.g. taking the union with any set of equal or smaller cardinality, taking the Cartesian prod... | 21 | https://mathoverflow.net/users/4832 | 64245 | 39,695 |
https://mathoverflow.net/questions/64185 | 3 | Hi everyone
I try to use GNS-construction to show every graded C\*-algebras can be faithfully represented on a graded Hilbert space.
If $A$ is a graded C\*-algebra with grading automorphism $\alpha$ of order 2, $\phi$ is a state on $A$. I think I need $\alpha (\phi(a))=\phi(a)$ for all $a\in A$ to get a grading on the... | https://mathoverflow.net/users/9401 | Graded $C^*$-algebras can be faithfully represented on a graded Hilbert space | In your case, if your algebra is separable, you can always produce a $\mathbb{Z}/2\mathbb{Z}$-invariant faithful state by averaging over the group: if $\phi$ is any faithful state, then $\psi(x) = \frac{1}{2}(\phi(x)+\phi(\alpha(x)))$ is $\mathbb{Z}/2\mathbb{Z}$-invariant (and, obviously, faithful).
Much more genera... | 3 | https://mathoverflow.net/users/12660 | 64251 | 39,698 |
https://mathoverflow.net/questions/64250 | 1 | Let $X$ be a compact complex n-fold . Then for every coherent sheaf $\mathfrak{F}$ on $X$ , and every holomorphic line bundle $L$ on $X$ , then the dimension of $H^0 (X,\mathfrak{F}\otimes\mathcal{O}\_X(L))$ does not depend on $L$ when dim Supp$\mathfrak{F}=0$ .
| https://mathoverflow.net/users/4437 | how to prove the following fact in sheaf cohomology ? | $\dim\mathrm{Supp}\\, \mathfrak F=0$ implies that $\mathfrak F\otimes \mathscr O\_X(L)\simeq \mathfrak F$ and hence its cohomology is independent of $L$.
| 9 | https://mathoverflow.net/users/10076 | 64253 | 39,700 |
https://mathoverflow.net/questions/64239 | 12 | Hello,
I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.
Let F be a p-adic field. Then (from what I understand) $C\_c^{\infty}(F)$ is the set of functions $f: F \to \mathbb{C}$ such that $f$ is locally constant and $f$ has compact support. Is the locally... | https://mathoverflow.net/users/14574 | Locally constant functions with compact support = smooth ? | A function $f : F \to \mathbb{C}$ is said to be smooth provided that $f$ is invariant under translation by some open subgroup of $F$ (hence by some compact open subgroup). Notice that when $f$ has compact support, this is equivalent to requiring that $f$ be locally constant. I don't have time to cook up a counterexampl... | 12 | https://mathoverflow.net/users/3544 | 64256 | 39,702 |
https://mathoverflow.net/questions/64255 | 1 | Hello!
Let $x\_1\ge x\_2\ge ... \ge x\_n>0$. Here are two optimization problems:
**P1: Maximize $\Pi \_{i=1}^n x\_i$ subject to $\sum \_{i=1}^n x\_i =K$.**
**P2: Minimize $\sum \_{i=1}^n (x\_i-K/n)^2$.**
As $\sum \_{i=1}^n x\_i =K$, $K/n$ actually is the mean value of $x\_i$. For Problem 1, the maximizer is $x\_1... | https://mathoverflow.net/users/12734 | Prove two optimization problems are equivalent | **NEW ANSWER**
Now that the problem has been clarified somewhat, under the particular constraint mentioned here, (but not others of a similar nature), the two problems do have the same solution. Let $K,n,\alpha$ be positive with $\alpha \gt \frac{K}{n}$ and consider these optimization problems over non-negative $n$-t... | 2 | https://mathoverflow.net/users/8008 | 64263 | 39,708 |
https://mathoverflow.net/questions/64268 | 3 | What are examples of coherent sheaves $\mathfrak{F}$ on a compact complex $n$-fold with $\dim \operatorname{Supp} \mathfrak{F}=p$ , where $0\leq p \leq n$ ? And how can they be described in local coordinates under the equivalence with vector bundles?
| https://mathoverflow.net/users/4437 | who can give me a example of coherent sheaf | Coherent sheaves are much more general than vector bundles. A vector bundle corresponds to a *locally free* sheaf; the support of such a sheaf is the whole variety.
If $X$ is your variety, of dimension $n$, and $Y \subset X$ is some closed subvariety of $X$, and $\mathcal{F}$ is some sheaf on $Y$, then there is a coh... | 5 | https://mathoverflow.net/users/2481 | 64275 | 39,713 |
https://mathoverflow.net/questions/64266 | 3 | Is it possible to know whether the given line bundle over compact complex manifold or projective variety is trivial or not from its sheaf cohomology data?
I found this question when I trying to solve the exercise problem in voisin's book Hodge theory and complex algebraic geometry 1
(I haven't solve that problem ye... | https://mathoverflow.net/users/14945 | Triviality of line bundle over complex manifold | There is one useful fact that might by relevant to you. A line bundle $L$ and a complex projective variety $X$ is trivial if and only if $H^0(X,L) \neq 0$ and $H^0(X,L^{-1})\neq0$.
Indeed one implication is clear, and for the other choose two non-zero elements $s \in H^0(X,L)$ and $s' \in H^0(X,L^{-1})$. Then $ss' \... | 2 | https://mathoverflow.net/users/5101 | 64276 | 39,714 |
https://mathoverflow.net/questions/58632 | 4 | Hi,
When Aetkin linear model is used, problem holder has to provide weight matrix which is defined as $\Sigma^{-1/2}$. As far as the covariance matrix is always positive-defined the raising to the power $1/2$ is just decomposition like $L^T L$ which is performed by Cholesky technique. In my case the covariance matrix... | https://mathoverflow.net/users/13688 | how to deal with bad-scaled covariance matrix? | If I understand your question properly, you should find a correlation matrix which is close, in some sense, to a given matrix which may have negative eigenvalues. This problem arises often on practice, and there are quite a few papers devoted to it. For example:
N.J. Higham, Computing the nearest correlation matrix -... | 4 | https://mathoverflow.net/users/1223 | 64278 | 39,715 |
https://mathoverflow.net/questions/64272 | 6 | Metrizability and complete regularity are topological properties that are, in a sense, different from the Hausdorff condition because they are not defined purely in the terms of the open sets, but rather using some external object, namely $\mathbb R$. Now a space is metrizable if it has the weak topology induced by a f... | https://mathoverflow.net/users/4903 | Characterization of Tychonoff spaces in terms of open sets | A $T\_1$ space $X$ is completely regular if and only if it has a basis $\mathcal{B}$ such that
1. For every $x \in X$ and every $U \in \mathcal{B}$ that contains $x$, there exists $V \in \mathcal{B}$ such that $x \notin V$ and $U \cup V = X$.
2. For any $U, V \in \mathcal{B}$ satisfying $U \cup V = X$, there exist $U... | 9 | https://mathoverflow.net/users/12547 | 64284 | 39,719 |
https://mathoverflow.net/questions/62981 | 22 | **Introduction.** I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction of the "ultra"ness.
**An example.** Given a free (ultra)filter $F$ on $\omega$, **Grigorieff forcing** is defined a... | https://mathoverflow.net/users/7281 | How "much" does (Grigorieff) forcing destroy an ultrafilter? | Here's a proof that, if $F$ is an ultrafilter and $g$ is $F$-Grigorieff-generic, then $F\cup\{g\}$ does not generate an ultrafilter in the extension. Define a real $x:\omega\to2$ (also viewed as $x\subseteq\omega$ as usual) by letting $x(n)=\sum\_{k=0}^ng(k)$ modulo 2. (Technically, I should fix the obvious names for $... | 10 | https://mathoverflow.net/users/6794 | 64287 | 39,721 |
https://mathoverflow.net/questions/64249 | 6 | Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a topological space. Equivalently, it is a topological stack arising from an etale topological groupoid. It is well known th... | https://mathoverflow.net/users/4528 | Automorphism groups and etale topological stacks | I don't think this is true. Let $X$ be the quotient of the action of $\mathbb Q$ on $\mathbb R$ by translation. This is a sheaf, and its automorphism groups are trivial. Suppose that there exist a local homeomorphism $U \to X$, where $U$ is non-empty a topological space. Let $V \to U$ be the pullback to $U$ of the $\ma... | 8 | https://mathoverflow.net/users/4790 | 64288 | 39,722 |
https://mathoverflow.net/questions/64277 | 7 | It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group scheme G/k. My question is do we have the similar statement for quasi-coherent D-modules? I hope that the category of qua... | https://mathoverflow.net/users/11964 | For quasi-coherent D-Modules | Hi Lei, I think your second statement is false, even in char. $0$: Let $X=\mathbb{A}\_k^1$ for some field $k$, and $U:=X\setminus \{0\}$. Denote the open immersion by $i$. Then $i\_\*\mathcal{O}\_U$ is $\mathcal{O}\_X$-quasi-coherent and not coherent. Consider the canonical connection on $i\_\*\mathcal{O}\_U$: If $x$ i... | 4 | https://mathoverflow.net/users/259 | 64289 | 39,723 |
https://mathoverflow.net/questions/64298 | 5 | Denote by $\mathbf{Ban}$ the category of Banach spaces and bounded linear maps and by $\mathbf{Banc}$ the subcategory of Banach spaces and linear contractions. The isomorphisms of $\mathbf{Ban}$ are the bounded linear bijections (open mapping theorem) and in $\mathbf{Banc}$ they are the isometric (linear) isomorphisms ... | https://mathoverflow.net/users/2562 | Short five lemma in Banach spaces | How about considering $\mathbb R^2$ with $l^1$ and $l^2$ norms. The restriction to the $x$-axis is an isometry. The quotient onto the $y$-axis, also an isometry. But not an isometry on the whole space.
| 11 | https://mathoverflow.net/users/454 | 64300 | 39,729 |
https://mathoverflow.net/questions/64302 | 10 | What is the best method for 1D numeric differentiation? Something as glorious as Gaussian quadrature for numeric integration.
Maybe differential quadrature is such a method? What is its accuracy?
I'm well aware that it is really easy to have symbolic differentiation in the program (automatic differentiation or trul... | https://mathoverflow.net/users/14251 | Numerical Differentiation. What is the best method? | If your function is badly behaved (e.g. noisy, very oscillatory), no method will perform properly (differentiation is numerically very *unstable*). That being said, for "nice functions", I have good experience with polynomial (Richardson) extrapolation methods. [This paper](http://dx.doi.org/10.1007/BF02166671) and [th... | 17 | https://mathoverflow.net/users/7934 | 64306 | 39,732 |
https://mathoverflow.net/questions/64308 | 0 | Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but
I simply carry over the term since the gist is the same: You have a set of diagrams thought to be identical outside an area and differ... | https://mathoverflow.net/users/11504 | "Skein" equations sets that can reduce any graph | The short answer is you can't. The long answer is that you are working with a presentation. Then a standard approach is that you should first make your relations confluent. You can make them locally confluent by the Knuth-Bendix aka Groebner basis algorithm. This may not terminate so you need some luck here. Once you h... | 1 | https://mathoverflow.net/users/3992 | 64311 | 39,735 |
https://mathoverflow.net/questions/64261 | 11 | One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details.
A look at the [DLMF](http://dlmf.nist.gov/) says that "the" multidimensional theta function is the [Riemann theta function](http://dlmf.nist.gov/21.... | https://mathoverflow.net/users/7934 | What's the difference between a Riemann theta and a Siegel theta function? | It's the same function. I checked with Maple 15:
$r := RiemannTheta([0.5+I, 2 I], Matrix(2, 2, [[I, 0], [0, I]]), [])$
$evalf(r)$
$-6.586149971\*10^6-2.132900065\*10^{-8} I$
Mathematica 8 gives
$N[SiegelTheta[\{\{I, 0\}, \{0, I\}\}, \{0.5 + I, 2 I\}]]$
$-6.58615\*10^6 - 8.06571\*10^{-10} I$
as Deconinck e... | 12 | https://mathoverflow.net/users/11260 | 64319 | 39,741 |
https://mathoverflow.net/questions/64323 | 18 | There is a commonly-encountered-but-wrong rule of thumb that says something like
>
> If a probability distribution is positively skewed, its mean is greater than its median.
>
>
>
(You sometimes also see it phrased in terms of mean and mode). **I'm looking for a nice counterexample to this rule.** What do I me... | https://mathoverflow.net/users/1256 | Looking for an appealing counterexample in probability | You are looking for random variables $X$ of median $m(X)$ such that
$$
E((X-E(X))^3)>0\quad\mbox{and}\quad E(X)< m(X).
$$
Assume there exists two nonnegative random variables $Y$ and $Z$ and an independent Bernoulli sign $B=\pm1$ with $E(B)=0$ such that $X=Y$ on $[B=1]$ and $X=-Z$ on $[B=-1]$.
Then $m(X)=0$ and $E(... | 12 | https://mathoverflow.net/users/4661 | 64337 | 39,752 |
https://mathoverflow.net/questions/64340 | 7 | The smallest two perfect numbers $n=6$ and $m=28$ satisfy
$$
\frac{m}{n+1} = 2^k
$$
with $k=2.$
Question: Are there more pairs of perfect numbers $n,m$ with $n < m$
and such that
$$
\frac{m}{n+1} = 2^k
$$
for some positive integer $k>0.$
Observe that the perfect number $n$ , the smallest of $n,m$
may be also ... | https://mathoverflow.net/users/11016 | Perfect numbers $n$ such that $2^k(n+1)$ is also perfect | If $m$ is odd, it's clearly impossible.
If $m$ is even and $n$ is odd, I don't know.
So suppose $m$, $n$ both even. Then $m=2^{r-1}p$ where $p=2^r-1$ is prime, and $n=2^{s-1}q$ where $q=2^s-1$ is prime, and $s\lt r$.
The equation becomes $$2^k(n+1)=2^k(2^{s-1}q+1)=2^{k+s-1}q+2^k=2^{r-1}p$$ Now $2^k$ divides th... | 3 | https://mathoverflow.net/users/3684 | 64341 | 39,755 |
https://mathoverflow.net/questions/64346 | 2 | I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions.
I am attempting a translation (with explanatory notes) of B. L. van der Waerden's 1932 papper "Stetigkeitssätze für halbeinfache Liesch... | https://mathoverflow.net/users/14510 | Meaning of "Compact" in 1932 Paper by van der Waerden "Continuity Theorem for Semisimple Lie Groups". | Any compact manifold is sequentially compact, if by "manifold" we understand "second-countable Hausdorff etc." This is because such spaces are metrizable by Urysohn's metrization theorem, and for metrizable spaces compactness and sequential compactness are equivalent. Wikipedia tells me this result was published in 192... | 2 | https://mathoverflow.net/users/290 | 64350 | 39,759 |
https://mathoverflow.net/questions/64294 | 10 | Let $Z \to X$ be a closed immersion of schemes. Is it true that for every morphism $Z \to Y$, the pushout $X \cup\_Z Y$ in the category of schemes exists? If yes, a) does it turn out to be simply sthe pushout in the category of locally ringed spaces, b) is the natural morphism $Y \to X \cup\_Z Y$ a closed immersion?
... | https://mathoverflow.net/users/2841 | Gluing along closed subschemes | In the positive direction, see D. Ferrand, "Conducteur, descente et pincement", Bull. SMF 131 (4), 2003, 553-585, especially Th. 5.4: the answer to all questions is yes if $Z\to Y$ is finite *and* every finite set in $X$ (resp. $Y$) is contained in an affine open subset.
More generally, assuming $Z\to Y$ affine, Theo... | 6 | https://mathoverflow.net/users/7666 | 64357 | 39,765 |
https://mathoverflow.net/questions/64359 | 5 | Hi there,
Here is a part of the book of Murty "an introduction to Sieve methods and applications"
page 36
"At this point we invoke some algebraic number theory let $K=Q(\theta)$ where $\theta$ is the solution of the polynomial $f(x)$. The ring of integers $O\_{k}$ of K is a Dedekind Domain. it's a classical theorem... | https://mathoverflow.net/users/4486 | PNT for number fields. | This statement is true for all $p$ not dividing $disc(f)$, as you say. Moreover you can write down exactly what the primes dividing $p$ are: they're the ideals $(p, g(\theta))$ for each irreducible factor $g$ of $f$. The norm of such a prime is the degree of $g$, so the number of degree 1 primes is the number of roots ... | 3 | https://mathoverflow.net/users/2481 | 64360 | 39,767 |
https://mathoverflow.net/questions/63971 | 8 | Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}\_F$ and $\mathcal{O}\_K$ be the associated rings of integers, and let $n\geq 1$.
>
> When is
> $$
> K\_{2n-1}(\mathcal{O}\_F)^G \cong K\_{2n-1}(\mathcal{O}\_K)?
> $$
>
>
>
It would be good enough for me to have this on th... | https://mathoverflow.net/users/35416 | Galois descent for K-groups (or for étale cohomology groups) | Sorry to answer my own question, but according to [these notes](http://www.ictp.it/~pub_off/lectures/lns015/Kolster/Kolster_Final.pdf) by Manfred Kolster, Proposition 2.9, the étale cohomology groups in fact always satisfy Galois descent. Should have googled harder before asking.
| 3 | https://mathoverflow.net/users/35416 | 64373 | 39,774 |
https://mathoverflow.net/questions/64370 | 37 | What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?
---
The only simple example known to me:
------------------------------------
In Jacobson's Basic Algebra (vol. 1), Section 2.8, there is an exercise that goes as follows:
Let $u=\... | https://mathoverflow.net/users/9672 | Simplest examples of rings that are not isomorphic to their opposites | Here's a factory for making examples. If $\Gamma$ is a quiver, and $k$ a field, then we get a quiver algebra $k\Gamma$. If $\Gamma$ has no oriented cycles, we can recover $\Gamma$ from $k\Gamma$ by taking the Ext-construction. Also, the opposite algebra of a quiver algebra is obtained by reversing all the arrows in the... | 42 | https://mathoverflow.net/users/14901 | 64384 | 39,781 |
https://mathoverflow.net/questions/63558 | 8 | After I posted [this question](https://mathoverflow.net/questions/56316/measurability-of-integrated-functions), a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main point of my previous question, after I g... | https://mathoverflow.net/users/11618 | Universally measurable sets and weak topology | After some time I found the solution to **Question 1** in the excellent book:
"Stochastic Optimal Control: The Discrete-Time Case", by Dimitri P. Bertsekas and Steven E. Shreve, freely available at [link](http://web.mit.edu/dimitrib/www/soc.html)
The answer is YES, see Corollary 7.46.1 page 177, Chapter 7.
For wh... | 6 | https://mathoverflow.net/users/11618 | 64389 | 39,785 |
https://mathoverflow.net/questions/64390 | 0 | (I'm not sure how to tag this; I'm tagging it math.CO because that's where it arose, but math.FA or something might be appropriate as well. Feel free to edit or comment on what this should be.)
I have a polynomial in a bunch of variables $P(x\_1, x\_2, \ldots, x\_n, y\_1, y\_2, \ldots, y\_m)$. I know that if we evalu... | https://mathoverflow.net/users/382 | Constructing a monotonic function | The answer is no. Is suffices to construct a polynomial such that these points are local minima. An example when $n=1$ is
$$x^2(y-1)^2(\epsilon+y^2(x-1)^2)$$
with $0<\epsilon$ small.
| 3 | https://mathoverflow.net/users/8799 | 64392 | 39,786 |
https://mathoverflow.net/questions/64399 | 10 | Suppose $A$ is a Noetherian (not necessarily local) ring and $\mathfrak{m}\subset A$ a maximal ideal. Then is it true that $$\hat{A}\_{\hat{\mathfrak{m}}}=\widehat{A \_{\mathfrak{m}}},$$ where hats denote completion and subscripts denote localization? If one uses superscripts to denote completion it would be
$$(A^{\m... | https://mathoverflow.net/users/5292 | Does completion commute with localization? | It is true. $(\widehat{A}, \widehat{\mathfrak{m}})$ is a Noetherian local ring so your left hand side could be simplified replacing it by $\widehat{A}$. Now let's use the definitions: $\widehat{A} = \varprojlim A/\mathfrak{m}^n$, whereas $\widehat{A\_{\mathfrak{m}}} = \varprojlim A\_{\mathfrak{m}}/(\mathfrak{m}A\_{\mat... | 25 | https://mathoverflow.net/users/5498 | 64406 | 39,791 |
https://mathoverflow.net/questions/64381 | 6 | I have come across an integral of the form
$$\int\_{b}^{a}\cdots\int\_{b}^{a} \left( \sum\_{i=1}^{n}x\_i\right)^mdx\_1d x\_2\dots dx\_n.$$
I have a solution that makes use of the [partition function](http://en.wikipedia.org/wiki/Partition_%28number_theory%29), but I feel there should be a much nicer solution and I'm su... | https://mathoverflow.net/users/5378 | Integrating the multinomial over a hypercube | We want the coefficient of $t^m/m!$ in
$$ \int\_b^a\cdots \int\_b^a e^{(x\_1+\cdots+x\_n)t}dx\_1\dots dx\_n =
\left(\frac{e^{at}-e^{bt}}{t}\right)^n. $$
Expanding by the binomial theorem and then taking the coefficient of $t^m/m!$ from each term will give a formula.
| 11 | https://mathoverflow.net/users/2807 | 64409 | 39,793 |
https://mathoverflow.net/questions/62994 | 9 | I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation [here](http://www.uweb.ucsb.edu/~criedel/Old_MathOverflow_question_2011-4-26.htm).
Background
----------
A simple consequence of the singular value decomposition is that any vector $v$ in a vector space $V$ forme... | https://mathoverflow.net/users/5789 | Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? | Vague question:
The hierarchical higher order SVD would lead you to the canonical form for the GHZ state. When you have $H\_1 \otimes H\_2 \otimes \ldots \otimes H\_n$ you basically first Schmidt decompose your state $\psi$ relative to $H\_1 \otimes (H\_2 \otimes \ldots \otimes H\_N)$ and then continue like this unti... | 3 | https://mathoverflow.net/users/14976 | 64415 | 39,798 |
https://mathoverflow.net/questions/64419 | 18 | Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex representation of $G$ and if $H$ is the Hecke algebra of locally constant complex-valued functions on $G$ with compact su... | https://mathoverflow.net/users/1384 | Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic. | First, let us formulate the theorem of Harish-Chandra in a little more precise manner:
it is a priori obvious that the character of $V$ is well-defined as a distribution. Now the theorem says that this distribution is given by a locally $L^1$-function which is well defined and is locally constant on an open dense subse... | 13 | https://mathoverflow.net/users/3891 | 64421 | 39,802 |
https://mathoverflow.net/questions/64452 | 10 | I work entirely over a field of characteristic $0$, in case it matters.
Recall that a *Poisson algebra* is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a Lie bracket, i.e. it satisfies a Jacobi identity, and (2) a derivation in each variable. Or, maybe even better is th... | https://mathoverflow.net/users/78 | In the dictionary between Poisson and Quantum, what corresponds to Coisotropic? | Concerning your first question I have a couple of suggestions: first coisotropic is in some sense the best we can have in a truely Poisson situation: there is nothing like lagrangian (unfortunately).
As you already said, lagrangian sometimes is associated to pure states in the quantum regime, here the main argument i... | 8 | https://mathoverflow.net/users/12482 | 64478 | 39,831 |
https://mathoverflow.net/questions/64477 | 11 | (I am cross-posting this from math.SE as it seems to be slightly over the top for that site.)
I saw in the class the theorem:
Suppose $X$ is a separable metric space, and $Y$ is a polish space (metric, separable and complete) then there exists a $G\subseteq X\times Y$ which is open and has the property:
For all $... | https://mathoverflow.net/users/7206 | Universal sets in metric spaces | The possibly unsatisfying answer to your question is "sometimes." I will instead discuss the obviously equivalent question about universal closed subsets (it will let me use more standard notation later). Moreover, I will focus on the special case that $X$ and $Y$ are both Polish, since that has been examined more in t... | 10 | https://mathoverflow.net/users/14913 | 64488 | 39,836 |
https://mathoverflow.net/questions/64443 | 20 | Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that $p\left(v\right)=p\left(F\left(v\right)\right)$ for all $v \in \mathbb{R}^n$. B... | https://mathoverflow.net/users/7392 | Which norms have rich isometry groups? | As an answer to (1), consider any compact subgroup $G\subset O(n)$ and look at the $G$-invariant functions on $\mathbb{R}^n$ that have homogeneity $1$. As long as $G$ does not act transitively on the space of lines in $\mathbb{R}^n$, that set of functions will be properly larger than the $O(n)$-invariant functions of h... | 14 | https://mathoverflow.net/users/13972 | 64494 | 39,840 |
https://mathoverflow.net/questions/64476 | 0 | Let $x=(x\_1x\_2...x\_n)$ be a binary sequence of length $n$. The Varshamov-Tenengolts code $VT\_0(n)$ consists of all binary vectors $(x\_1, . . . , x\_n)$ satisfying $\Sigma\_{i=1}^n i\*x\_i \equiv0 \pmod {n+1} $.
Prove that $\forall$ $x,y \in VT\_0(n)$ which has equal hamming weight the Hamming distance between $x... | https://mathoverflow.net/users/15000 | Fixed Hamming distance property of binary deletion correcting codes | Take $n=11$. 11000000100 and 00111000000 are both in the code and they both have Hamming weight 3 but the Hamming distance between them is 6.
| 3 | https://mathoverflow.net/users/3684 | 64496 | 39,842 |
https://mathoverflow.net/questions/64484 | 3 | Let $A \subseteq \mathbb{R}^2$ be closed and connected, and for any $c \in \mathbb{R}$, let $A\_c \subseteq A$ be a closed and connected subset of $A$, s.t. for $c \neq d$ we have $A\_c \cap A\_d = \emptyset$.
If the Hausdorff-dimension of each $A\_c$ is at least 1, what lower bound can be given for the Hausdorff-dim... | https://mathoverflow.net/users/15002 | Hausdorff-dimension of connected closed subsets of R^2 | *As Joel pointed out, the MO system works better with answers (or hints) as answers rather than as comments. Even though Joel did not suggest giving a more complete answer, I will do so.*
Let $C$ be a Cantor set with Hausdorff-dimension $0$. Then $A = (C \times \mathbb{R}) \cup ([0,1]\times\{0\})$ has Hausdorff-dimen... | 4 | https://mathoverflow.net/users/11716 | 64499 | 39,843 |
https://mathoverflow.net/questions/64501 | 11 | I would like to get a reference where I can learn about the theory of projective representations of finite groups over the complex numbers (or over any field K such that the order of the given group under study is invertible in K). And how one can relate this to character theory for linear representations (with which I... | https://mathoverflow.net/users/4800 | Reference request for projective representations of finite groups over a non-problematic field | Many textbooks cover this material. For example, Curtis and Reiner, Representation Theory of Finite Groups and Associative Alegbras, Wiley,1962. Projective representations of finite groups (in the sense of Schur) are just genuine linear representations of central extensions. They often arise
in Clifford theory, which c... | 13 | https://mathoverflow.net/users/14450 | 64502 | 39,845 |
https://mathoverflow.net/questions/64511 | 19 | [Kirszbraun theorem](https://en.wikipedia.org/wiki/Kirszbraun_theorem) states that if $U$ is a subset of some Hilbert space $H\_1$, and $H\_2$ is another Hilbert space, and $f : U \to H\_2$ is a Lipschitz-continuous map, then $f$ can be extended to a Lipschitz function on the whole space $H\_1$ with the same Lipschitz ... | https://mathoverflow.net/users/3736 | Explicit extension of Lipschitz function (Kirszbraun theorem) | I like a recent proof by Akopyan and Tarasov:
A. V. Akopyan, A. S. Tarasov, "A constructive proof of Kirszbraun's
theorem"(Russian), Mat. Zametki 84 (2008), no. 5, 781--784;
translation in Math. Notes 84 (2008), no. 5-6, 725–728; MR2500644.
I could not find this paper in the open web, but there is a copy behind a
p... | 10 | https://mathoverflow.net/users/4354 | 64515 | 39,851 |
https://mathoverflow.net/questions/64512 | 7 | Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z\_1,\ldots,Z\_k\subseteq X$. Let $\mathcal{I}\_i$ be the ideal sheaf of $Z\_i$ and $\mathcal{I}:=\mathcal{I}\_1\cdots\mathcal{I}\_k$ the product. My question is whether $\mathcal{I}$ is the ideal sheaf of the uni... | https://mathoverflow.net/users/9947 | Products of Ideal Sheaves and Union of irreducible Subvarieties | For each $i$ we have a short exact sequence
$$
0 \to I\_i \to O\_X \to O\_{Z\_i} \to 0.
$$
Let us think about it as about a resolution of $I\_i$. Then it follows that
$$
I\_1 \otimes^L I\_2\otimes^L \dots \otimes^L I\_k =
(O\_X \to O\_{Z\_1}) \otimes^L (O\_X \to O\_{Z\_2}) \otimes^L \dots \otimes^L (O\_X \to O\_{Z\_k... | 6 | https://mathoverflow.net/users/4428 | 64519 | 39,852 |
https://mathoverflow.net/questions/64498 | 9 | I'm reading Daniel Quillen's paper "Homology of commutative rings," in which he proves:
A finitely presented morphism of rings $A \to B$ is
1. Formally etale iff $L\_{B/A}$ (this denotes the cotangent complex) is homotopy-equivalent to zero
2. Formally smooth iff $\Omega\_{B/A}$ is projective and $L\_{B/A}$ is hom... | https://mathoverflow.net/users/344 | Formally smooth morphisms, the cotangent complex, and an extension of the conormal sequence | We deduce this from the Jacobi-Zariski exact sequence as follows:
Some notation first: Let $H\_i(A,B,W)$ where B is an A-algebra and W is a B-module denote $\pi\_i(L\_{B/A}\otimes\_B W)$.
Then given an $A$-algebra $B$, a $B$-algebra $C$, and a $C$-module $W$, we have a long-exact sequence, called the Jacobi-Zarisk... | 6 | https://mathoverflow.net/users/1353 | 64529 | 39,857 |
https://mathoverflow.net/questions/64525 | 6 | This game looks like bridge, but 1- there are only two players Alice and Bob, 2- there is only one suit, whose cards are numbered $1, 2,\ldots,2n$. One deals each player $n$ cards. Therefore Alice knows Bob's cards and conversely; once the cards are dealt, there is no randomness.
Alice play a card, then Bob. The high... | https://mathoverflow.net/users/8799 | Bridge game with only one suit: strategy | I believe the game you describe is two-person single suit whist and was solved by Johan Wastlund in [this paper](http://www.combinatorics.org/Volume_12/PDF/v12i1r43.pdf).
| 19 | https://mathoverflow.net/users/422 | 64530 | 39,858 |
https://mathoverflow.net/questions/64520 | 13 | I'm looking at Kahler geometry at the moment and admiring how it manages to do so much with clean global algebraic arguments. One of the big exceptions to all this, however, is the proof of the Kahler identities
$$
[\Lambda,\overline{\partial}]=-i \partial^\ast, ~~~~~~
[\Lambda,\partial]=-i \overline{\partial}^\ast.
$... | https://mathoverflow.net/users/1648 | Global Algebraic Proof of the Kahler Identities? | I don't know of a proof that would really be characterized as global, and if I saw one I would immediately try to figure out how it's really local. There is, however a proof along different lines than that in GH or Voison, and seems more enlightening to me. Huybrechts in his Complex Geometry book gives one that is more... | 12 | https://mathoverflow.net/users/12678 | 64534 | 39,861 |
https://mathoverflow.net/questions/63789 | 27 | Let $(S\_n)\_{n=1}^{\infty}$ be a standard random walk with $S\_n = \sum\_{i=1}^n X\_i$ and $\mathbb{P}(X\_i = \pm 1) = \frac{1}{2}$. Let $\alpha \in \mathbb{R}$ be some constant. I would like to know the value of
$$\mathcal{P}(\alpha) := \mathbb{P}\left(\exists \ n \in \mathbb{N}: S\_n > \alpha n\right)$$
In other... | https://mathoverflow.net/users/11259 | Probability of a Random Walk crossing a straight line | I wrote a Maple program in 2003 which computes $P(\alpha)$ (both the "upper" and "lower" values) for a given rational number $\alpha$, much in line with the method described by Johan. I think I was able to compute it for all $\alpha=a/b$ for integers $1\le a\lt b\le 300$ in a couple of hours, so the program could proba... | 10 | https://mathoverflow.net/users/1304 | 64539 | 39,865 |
https://mathoverflow.net/questions/64536 | 8 | Let $\psi(x)=\sum\_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int\_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. Originally, this integral came up as a particular part in a particular case for a a formula for a summatory function I... | https://mathoverflow.net/users/12176 | Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$ | Consider $f(s):=\int\_1^{\infty}\frac{\psi(x)-x}{x^s}dx$, which converges for $Re(s)\geq2$. For $Re(s)>2$, we can separate the numerator and integrate by parts (using the Riemann-Stieltjes integral, for convenience) to get $f(s)=\frac{1}{s-1}\int\_1^{\infty}\frac{1}{x^{s-1}}d\psi(x)-\frac{1}{s-2}$. Now, $\frac{\zeta'}{... | 14 | https://mathoverflow.net/users/5263 | 64543 | 39,868 |
https://mathoverflow.net/questions/64549 | 1 | I am a programmer and have the following requirement.
We are trying to figure out the optimal way to ship widgets. Below is the scenario:
* We need to ship 1,000,000 widgets
* We have two different size boxes. A 300 widget size box and a 200 widget size box.
* The widgets are shipped to 2,000 individual distributor... | https://mathoverflow.net/users/15013 | Model for shipping widgets in an optimal way | Is the routing from factory to distributor to location fixed, or are you considering optimizing this too? In particular, are there costs associated with using a distributor? If the answers to one or both of these questions are yes, then you should take at the literature on "uncapacitated facility location problems."
... | 2 | https://mathoverflow.net/users/9022 | 64552 | 39,870 |
https://mathoverflow.net/questions/64551 | 12 | I am trying to find a reference for lower cohomology groups $H^i(G, \mathbb{Z}),$ for $i=1, 2, 3$ for lattices in higher rank (for example, $SL(n, \mathbb{Z}), Sp(2n, \mathbb{Z}),$ and possibly congruence subgroups thereof. There is the Borel periodicity formula, but that's over $\mathbb{Q}\dots$
| https://mathoverflow.net/users/11142 | Cohomology of lattice subgroups | I don't think this is known in complete generality. Here are some results. Everything is easy in rank 1, so I'll restrict myself to $SL\_n(\mathbb{Z})$ for $n \geq 3$ and
$Sp\_{2g}(\mathbb{Z})$ for $g \geq 2$. Denote the level $L$ congruence subgroups of $SL\_n(\mathbb{Z})$ and $Sp\_{2g}(\mathbb{Z})$ by $SL\_n(\mathbb{... | 12 | https://mathoverflow.net/users/317 | 64558 | 39,875 |
https://mathoverflow.net/questions/64556 | 15 | I am trying to find a formula for the following integral for non-negative integer $k$:
$$\int\_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$
My first thought was to use the formula $$\zeta(s)-\frac{1}{s-1}=1-s\int\_1^\infty u^{-s-1}\{u\}du$$ where $\{u\}$ refers to the fractional part. We can then take... | https://mathoverflow.net/users/12176 | Evaluating the integral $\int_{1}^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$ | Let $a\_k$ be the integral. Then
$$\begin{eqnarray\*}
\sum\_{k \ge 0} \frac{a\_k}{k!} t^k &=& \int\_1^{\infty} \frac{ \{ u \} }{u^2} e^{t \log u} \, du \\\
&=& \int\_1^{\infty} \{ u \} u^{t-2} \, du \\\
&=& \frac{1 - \zeta(1 - t) - \frac{1}{t}}{1 - t} \\\
&=& \frac{1}{1 - t} \left( 1 - \sum\_{n \ge 0} \frac{\gamm... | 21 | https://mathoverflow.net/users/290 | 64568 | 39,881 |
https://mathoverflow.net/questions/64356 | 1 | I have the following equation:
$Y = [ C \bullet R ] \times X \times[C^T \bullet R^T ]$
In my textbook, this conveniently rearranges to:
$Y = [ C \times X \times C^T ] \bullet [R \bullet R^T ]$
where $\times$ denotes matrix multiplication, $\bullet$ is Hadamard (pointwise multiplication), and $R^T$ is the transp... | https://mathoverflow.net/users/14963 | Rearrange equation involving matrix multiplication and Hadamard product | With these matrices the identity is true.
In order to prove it, notice that $R=DE$, where $D=\operatorname{diag}(\frac12,\frac1{\sqrt{10}},\frac12,\frac1{\sqrt{10}})$ and $E$ is the matrix of all ones. You can juggle around the diagonal factor $D$ (your proof shows that the "triple product identity" holds if $A$ is d... | 3 | https://mathoverflow.net/users/1898 | 64574 | 39,883 |
https://mathoverflow.net/questions/64461 | 14 | Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "basically" $\zeta(s)/\zeta(2s)$. However, there is the funny business at 2, and one derives the formula
$\sum D^{-s} = ... | https://mathoverflow.net/users/1050 | Is there an elegant algebraic proof of this formula for quadratic field discriminants? | The discriminant $D$ of a quadratic number field can be written uniquely as a product of prime discriminants, namely $-4$, $\pm 8$, and $p^\* = (-1)^{(p-1)/2}p$ for odd primes $p$.
The Dirichlet series for odd discriminants therefore simply is
$$ \sum\_{D \text{ odd}} |D|^{-s} = \prod\_{p \text{ odd}} (1 + p^{-s}), $$
... | 13 | https://mathoverflow.net/users/3503 | 64592 | 39,892 |
https://mathoverflow.net/questions/64591 | 5 | Assume we have an affine scheme $A$ that comes with a closed immersion into a smooth scheme $i \colon A \hookrightarrow M$, which is not necessarily affine.
Does there exist an affine open subscheme $j \colon V \hookrightarrow M$ such that $A$ already embeds in $V$?
Expressed in Diagrams, does there exist an affin... | https://mathoverflow.net/users/473 | Does a closed immersion of an affine scheme in a smooth scheme factor over an open affine subscheme? | Let me expand on my comment. Let $E$ be an elliptic curve and let $p$ be a point of $E$ of infinite order. Embed $E$ in $\mathbb{P}^2$ as a plane cubic using the linear system $3O$, where $O$ is the origin of the group law. Remove from $\mathbb{P}^2$ the image of the point $p$ and note that $E \setminus \{p\}$ is affin... | 16 | https://mathoverflow.net/users/4344 | 64594 | 39,894 |
https://mathoverflow.net/questions/64587 | 1 | A very basic question that seems to be unsolved: is the class of sofic/hyperlinear groups closed under semi-direct product?
In case of sofic groups the following restricted version is well-known: if $N$ is sofic and $H$ is amenable, then any semidirect product $N\rtimes H$ turns out to be sofic. Is the analogue true ... | https://mathoverflow.net/users/13809 | Sofic/hyperlinear groups | I think that this is known in the operator algebra commmunity, but it is also a consequence of the proof in the sofic case, which was obtained in
Elek, Gábor, Szabó, Endre, *On sofic groups*. J. Group Theory 9 (2006), no. 2, 161–171.
| 6 | https://mathoverflow.net/users/8176 | 64597 | 39,895 |
https://mathoverflow.net/questions/64596 | 3 | Let $S \subset 2^M$ be a family of subsets of a set $M$ with $|M| = n$. Call $S$ *min-k-intersecting* if for each pair of subsets the intersection has at most $k$ elements: $\forall A,B \in S: |A \cap B | \leq k$.
Question 1: What is the maximal $N(n,k) = |S|$ of such a min-k-intersecting family of subsets of a set w... | https://mathoverflow.net/users/15024 | minimal intersecting subsets | The answer (to question 1) is $\sum\_{j=0}^{k+1}{n\choose j}$ and is realised by all subsets containing at most $k+1$ elements of $M$. Two such distinct subsets have indeed at most $k$ elements in
common.
Let now $\mathcal S$ be a set of subsets in $M$ satisfying your requirement.
Suppose $A\in S$ is a subset of... | 6 | https://mathoverflow.net/users/4556 | 64598 | 39,896 |
https://mathoverflow.net/questions/64382 | 3 | I'm not sure if this is the right place to ask this question, but I'll ask it anyway, in the hope that some kindly Australian (true or honorary) is passing by and takes pity on me...
In *Fibrations in bicategories*, Street shows that V-profunctors are exactly the codiscrete cofibrations in the 2-category V-Cat (i.e. ... | https://mathoverflow.net/users/4262 | Comparing discrete fibrations and their duals | The fact that codiscrete cofibrations and discrete fibrations are equivalent is a very special exactness property of the 2-category Cat. I can't, off the top of my head, think of any other interesting examples. It's not true, for example in the 2-category 2-Cat of 2-categories, 2-functors, and 2-natural transformations... | 2 | https://mathoverflow.net/users/10862 | 64600 | 39,898 |
https://mathoverflow.net/questions/64595 | 7 | Let $V^k\subset \mathbb C^n$ be a sub variety, such that all its irreducible components have dimension $\ge k$. Is it true that $\mathbb C^n\setminus V^k$ has homotopy type of a CW complex of dimension $\le 2n-k-1$?
*Comments.* 1)This is true for $k=n-1$, since in this case $\mathbb C^n\setminus V^{n-1}$ is affine. ... | https://mathoverflow.net/users/13441 | Homotopy type of the complement to a subvariety of $\mathbb C^n$ | Take $n=4$ and let $V = \{ z\_1=z\_2=0 \} \cup \{ z\_3=z\_4=0 \}$. I claim that $\mathbb{C}^4 \setminus V$ is homotopic to $S^3 \times S^3$, which has nontrivial homology in degree $6$, contrary to your supposed bound, which is in degree $5$.
Note that $\mathbb{C}^4 \setminus V = \left( \mathbb{C}^2 \setminus \{ (0,0... | 8 | https://mathoverflow.net/users/297 | 64604 | 39,900 |
https://mathoverflow.net/questions/64578 | 0 | I've asked the [same question](https://math.stackexchange.com/questions/38238/hilbert-waring-theorem-using-the-sum-of-squares-function) at the Math Stack Exchange site, but I didn't have any luck there. So I'm posting the same question here.
Denote by $r\_{s,k}(x)$, the number of ways in which $x$ can be expressed a... | https://mathoverflow.net/users/7144 | The Hilbert-Waring theorem using the sum-of-squares function. | The negation of the statement
$$ \forall s\in \mathbb{N}\; \exists\: k \; \: \text{such that} \;r\_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N} $$
is not the statement
$$ \forall s\in \mathbb{N}\; \nexists\: k \; \: \text{such that} \;r\_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N} $$
but the statement
$$ \exists s\in \m... | 5 | https://mathoverflow.net/users/11919 | 64616 | 39,907 |
https://mathoverflow.net/questions/64610 | 22 | At the end of "Notes on Chapter 1" in the Preface to the Third Edition of *Sphere packings, lattices and groups*, Conway and Sloane write the following:
>
> Finally, we cannot resist calling attention to the remark of Frenkel, Lepowsky and Meurman, that **vertex operator algebras** (or conformal field theories) are... | https://mathoverflow.net/users/484 | Codes, lattices, vertex operator algebras | I think the analogy you describe cannot be made precise with our current technology. For example, the word "functor" doesn't seem to have made an appearance yet in this context.
If you have a code, there are methods to construct lattices using it, but some of the constructions (like the Leech lattice) require special... | 10 | https://mathoverflow.net/users/121 | 64628 | 39,915 |
https://mathoverflow.net/questions/64613 | 6 | Let $V$ be a subvariety of $\mathbb C^n$ with irreducible components of dimension >$0$. Is $H\_{2n-1}(\mathbb C^n\setminus V)=0$?
| https://mathoverflow.net/users/13441 | A bound on the top homology of a complement to a variety in $\mathbb C^n$ | It's certainly true, though I don't know the best way of proving it.
One way to see it is that if you take the one-point compactifications $V^+\subset(\mathbb{C}^n)^+=S^{2n}$, you get a connected space, since all components contain the point at infinity. The result then follows by Alexander duality in $S^{2n}$.
| 6 | https://mathoverflow.net/users/14901 | 64632 | 39,918 |
https://mathoverflow.net/questions/64619 | 16 | I thought of the following probability problem, which seems to have an answer of 1/e, and wonder if someone has an idea as to how to prove this.
Suppose a man has a bottle of vitamin pills and wishes to take a half pill per day. He selects a pill from the bottle at random. If it is a whole pill he cuts it in half, ta... | https://mathoverflow.net/users/15027 | Probability Problem Involving e | One way of solving the problem (probably not the easiest) is via the "differential equation method" (see eg the paper by Nick Wormald, entitled "The differential equation method for random graph processes and greedy algorithms", in Lectures on Approximation and Randomized Algorithms, pp. 73-155).
Suppose there are $n... | 12 | https://mathoverflow.net/users/9386 | 64638 | 39,924 |
https://mathoverflow.net/questions/64546 | 4 | Let $V^{2k}$ be a complex subvariety of dimension $2k$ (real dimension $4k$) in $\mathbb C^n$. Let $A$ be a complex $n-k$ dimensional plane in $\mathbb C^n$.
**Question.** Is it true that the inclusion $H\_{2n-2k-1}(A\setminus (V\cap A))\to H\_{2n-2k-1}(\mathbb C^n\setminus V)$ is injective?
We don't require $V^{2k... | https://mathoverflow.net/users/13441 | An analogue of Lefschetz hyperplane theorem for complements to subvarieties in $\mathbb C^n$ ? | Yes. Indeed all irreducible components of $V\cap A$ have positive dimension. So the map is injective, since $H\_{2n-2k-1}(A\setminus (V\cap A))=0$ as is shown in the answer to the following question:
[A bound on the top homology of a complement to a variety in $\mathbb C^n$](https://mathoverflow.net/questions/64613/a... | 1 | https://mathoverflow.net/users/13441 | 64654 | 39,934 |
https://mathoverflow.net/questions/64659 | 6 | Call a graph $G$ $n$-saturated if for every set $A$ of size $n$ of vertices and all $B\subseteq A$ there is a vertex $v\not\in A$ that forms an edge with all $w\in B$ and
does not form an edge with any $w\in A\setminus B$.
A countably infinite graph is isomorphic to the random graph iff it is $n$-saturated for all ... | https://mathoverflow.net/users/7743 | Finite graphs that realize all types over $n$-element sets | 1. If $G=G(m,\frac12$) then for any fixed set $A$ and subset $B$ and $v$ we have the probability that $v$ is connected to $B$ but not to $A\setminus B$ is $2^{-n}$. The probability that all $v$ fail to do so is $(1-2^{-n})^{m-n} \approx e^{m 2^-n}$. There are less than $(2m)^n$ such subsets $A$ and $B$, so union bound ... | 5 | https://mathoverflow.net/users/1061 | 64663 | 39,940 |
https://mathoverflow.net/questions/64656 | 0 | If I have a general d-dimension cubic form $C\_0(x)$ with coefficients in $\mathbb{R}$ is it possible to find a cubic form $C\_1(x)$ with coefficients in $\mathbb{R}$ such that for all $x\in\mathbb{R}^{d}$ the following are satisfied:
i) $C\_1(x)=Q(x)L(x)$ where $Q(x)$ and $L(x)$ are quadratic and linear forms respec... | https://mathoverflow.net/users/15031 | Is it possible to approximate a general cubic form by one which factorises? | I would expect not. The reason being that if this is anything like true (products of quadratic forms and and linear forms being dense in the ordinary "strong" or norm topology - presumably what you mean) then more would be true than that. So for a proof one could either write down an equation that the coefficients of s... | 0 | https://mathoverflow.net/users/6153 | 64670 | 39,946 |
https://mathoverflow.net/questions/64695 | 5 | Let $X=(V,E)$ be a finite, connected, regular graph with diameter $D$. Is it true that, for every $x\in V$, there exists $y\in V$ such that $d(x,y)=D$? (the answer is clearly yes if $X$ is vertex-transitive).
| https://mathoverflow.net/users/14497 | Realizing the diameter of a finite regular graph | [Counter-example http://www.freeimagehosting.net/uploads/9a4165cab4.png](http://www.freeimagehosting.net/uploads/9a4165cab4.png)
The diameter is 8, but 1 is centered with at most 5 as distance to every other.
| 6 | https://mathoverflow.net/users/14948 | 64700 | 39,970 |
https://mathoverflow.net/questions/64701 | 1 | I am trying to obtain an analytic estimate of this integral:
$\int\_0^1\frac{1}{\sqrt{x}}\exp\left(-a(x-x\_0)^2\right) dx$,
where $a\gg1$, $x\_0\in[0,1]$. Saddle-point approximation doesn't work due to infinite derivative of $1/\sqrt{x}$ at 0. Any tips on how to get a handle on this will be much appreciated.
| https://mathoverflow.net/users/1783 | Estimating an integral with a singularity at the interval's endpoint | Well, now you tell us that $x\_0$ depends on $a$. If $x\_0$ is of order $1/\sqrt{a}$, set $x\_0=y\_0/\sqrt{a}$, $x=y/\sqrt{a}$, and your integral becomes
$$a^{-1/4}\int\_0^{\sqrt{a}}{1\over\sqrt{y}}\exp(-(y-y\_0)^2)\,dy.$$
Up to exponentially small error, you can replace the upper bound of $\sqrt{a}$ by $\infty$.
| 1 | https://mathoverflow.net/users/12120 | 64712 | 39,976 |
https://mathoverflow.net/questions/64717 | 10 | On a manifold equipped with C^k atlas (with k>0) there is essentially one smooth structure compatible with the atlas. According to Wikipedia, this is a result due to Whitney. This is in stark contrast with a C^0 atlas, where there might exist many smooth structures or none at all.
I was wondering, what is the underly... | https://mathoverflow.net/users/15038 | Smooth structures compatible with a given C^1 structures | See my answer to [this question](https://mathoverflow.net/questions/58061/how-can-there-be-topological-4-manifolds-with-no-differentiable-structure/58087#58087).
Consider the space (in some appropriate sense ) of all invertible germs of $C^{\infty}$ maps from $\mathbb R^n$ to itself fixing the origin. This is homoto... | 15 | https://mathoverflow.net/users/6666 | 64722 | 39,983 |
https://mathoverflow.net/questions/64716 | 8 | In another post Ryan Budney has mentioned a "smooth proof" of the theorem of Reidemeister...
(@Ryan Budney:) Do you know if there is a book (or paper) containing such a proof? I am interested in and can't find one...
Thank you very much,
sincerely,
Johannes Renkl
| https://mathoverflow.net/users/15037 | Smooth proof of Reidemeister theorem | I don't believe it's written up anywhere.
edit: in the comments Charlie Frohman corrects me:
>
> [MR2128054](http://www.ams.org/mathscinet-getitem?mr=2128054) (2005m:57041)
> Roseman, Dennis(1-IA)
> Elementary moves for higher dimensional knots. (English summary)
> Fund. Math. 184 (2004), 291–310.
> 57R40 (5... | 13 | https://mathoverflow.net/users/1465 | 64723 | 39,984 |
https://mathoverflow.net/questions/64731 | 1 | Consider $a\_n$ a real valued sequence and define $D\_{1,1,1}(s)=\sum\_{n=1}^\infty \frac{a\_n}{n^s}$ which converges in some half plane $\Re s =c.$ Define $D\_{r,h,k}(s) = \sum\_{n=1}^\infty \frac{e^{2\pi i \frac{hrn}{k}} a\_n}{n^s}.$
Question: Suppose $D\_{1,1,1}(s)$ has an analytic continuation to a half plane $R$... | https://mathoverflow.net/users/12337 | Dirichlet Series Question | A counterexample is provided by $c=1$ and $a\_n=\sum\_{d\mid n}(-1)^d$.
Indeed,
$$ D\_{1,1,1}(s)=(2^{1-s}-1)\zeta(s)^2 $$
has a simple pole at $s=1$, but
$$ D\_{1,1,2}(s)=\sum\_{n=1}^\infty \frac{(-1)^na\_n}{n^s}=(1-2^{1-s}+2^{1-2s})\zeta(s)^2 $$
has a double pole at $s=1$.
**EDIT:** Daniel asked in a comment how I c... | 11 | https://mathoverflow.net/users/11919 | 64736 | 39,993 |
https://mathoverflow.net/questions/62924 | 4 | Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ of $h$, is a diffeomorphism of $D$ and homeomorphism of $\bar{D}$.
My question is : suppose $h:S^1\to S^1$ is a k-qua... | https://mathoverflow.net/users/6953 | Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$ | For the unit disk see: <http://poincare.matf.bg.ac.rs/~pavlovic/QC_DIF.PDF>
For the halp-plane: <http://poincare.matf.bg.ac.rs/~pavlovic/QC_DIF.PDF>
For a generalization to Jordan domains: <http://arxiv.org/abs/0910.4950> (see also my paper in MATH z)
| 2 | https://mathoverflow.net/users/15042 | 64753 | 40,006 |
https://mathoverflow.net/questions/64746 | 12 | This question is related to [Realizing the diameter of a finite regular graph](https://mathoverflow.net/questions/64695/realizing-the-diameter-of-a-finite-regular-graph)
Let $X=(V,E)$ be a finite, connected, regular graph of diameter $D$. Assume that, for every vertex $x\in V$, there exists some vertex $y\in V$ such ... | https://mathoverflow.net/users/14497 | "Antipodal" maps on regular graphs? | I don't know. The following is a near miss which might be useful.
Start with a hexagonal cycle path ABCDEFA. Duplicate point C to C' and connect C' to
B,C, and D. Similarly duplicate points E and F, and add edges EE', FF', and the 3 edges
to form the path DE'F'A. Then it has diameter 3, but the only point that is di... | 5 | https://mathoverflow.net/users/3402 | 64754 | 40,007 |
https://mathoverflow.net/questions/64649 | 18 | The *easier Waring problem* asks for the least number $v=v(k)$ such that every every integer is a sum of $v$ $k$'th powers *with signs*, i.e. every $n\in \mathbb{N}$ is of the form $$n=x\_1^k\pm x\_2^k\pm\dotsb\pm x\_v^k.$$
The problem is ``easier'' because unlike the usual Waring problem (without the signs) the exis... | https://mathoverflow.net/users/806 | Lower bounds on the easier Waring problem | As far as I am aware, nothing unconditional is known. The difficulty is that one has no obvious constraint on the size of the variables, so that the $x\_i$ could be arbitrarily large in terms of $n$ in a solution. The difficulty of ruling out solutions (when congruence conditions do not rule out solubility) is related ... | 19 | https://mathoverflow.net/users/15044 | 64765 | 40,014 |
https://mathoverflow.net/questions/64545 | 3 | In the original paper of Mehta-Seshadri, it seems like they treat the case of zero parabolic degree (i.e. they prove that zero parabolic degree stable parabolic bundles correspond to irreps of the fundamental group of the Riemann surface minus some points). But, as in the Narasimhan-Seshadri theorem, is the nonzero deg... | https://mathoverflow.net/users/3709 | Mehta-Seshadri and Parabolic bundles | I think what you want probably follows from Theoreme 2.5 of the paper:
Biquard, Olivier: Fibrés paraboliques stables et connexions singulières plates,
Bull. Soc. Math. France 119 (1991), no. 2, 231–257.
| 3 | https://mathoverflow.net/users/519 | 64772 | 40,019 |
https://mathoverflow.net/questions/64776 | 18 | Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is *never* flat (see Liu, example 4.3.5).
What happens if we suppose $X$ normal, and we take the normalization in a finite (separable) extension of the function field of $X$? Note t... | https://mathoverflow.net/users/7845 | Flatness of normalization | A characteristic zero example: Let $k$ be a field of characteristic zero (or anything not $2$.) Let $L$ be the field $k(x,y)$ and let $K$ be the subfield $k(x^2, xy, y^2)$, so $[L:K]=2$. Let $S \subset L$ be the ring $k[x,y]$, and let $R = S \cap K = k[x^2, xy, y^2]$.
Then $S$ is the normalization of $R$ in $L$. I c... | 9 | https://mathoverflow.net/users/297 | 64786 | 40,025 |
https://mathoverflow.net/questions/64773 | 4 | Given an algorithm A that consumes an infinite sequence of bits and can terminate at any point, it's straightforward to prove that either A's running time is uniformly bounded or there's an infinite bit sequence S that makes A run forever. I'm interested in the computability of S. When S is unique, I believe I know an ... | https://mathoverflow.net/users/13669 | Finding inputs that make an algorithm run forever | There is such an algorithm, for which one cannot find a computable $S$ on which it runs forever.
There is a computable tree with no computable infinite branch. See the Wikipedia entry on [König's lemma](http://en.wikipedia.org/wiki/K%C3%B6nig%27s_lemma#computability_aspects); the existence of such trees shows that th... | 7 | https://mathoverflow.net/users/1946 | 64792 | 40,030 |
https://mathoverflow.net/questions/64777 | 3 | At the very beginning of Feferman's *Arithmetization of metamathematics in a general setting* it can be read:
>
> The method of arithmetization, as developed by Gödel[10], exploits the possibility of defining within a formal theory $\mathcal{T}$, or in arithmetical theories closely related to $\mathcal{T}$, various... | https://mathoverflow.net/users/6466 | Feferman's extensional and intensional applications of the method of arithmetization | As indicated by Feferman, the key distinction is between two sorts of arithmetical definitions of certain concepts (like "being the Gödel number of a theorem of a given theory"). Suppose I have some set $S$ of integers in mind and I propose a formal definition of it in some theory $T$, i.e., a formula $\phi(x)$ intende... | 6 | https://mathoverflow.net/users/6794 | 64796 | 40,031 |
https://mathoverflow.net/questions/64770 | 6 | Let $G=(V,E)$ be a finite simple $k-$regular graph ($k\geq 1$). Does $G$ necessarily
contain a subset $E'\subset E$ of edges such that only isolated edges and cycles occur as connected components in $(V,E')$?
(The answer is easily yes for $k=1,2$.)
A counterexample would easily give a counterexample to question ["A... | https://mathoverflow.net/users/4556 | Existence of a nice subset of edges in $k-$regular simple graphs? | It seems like such a subset should always exist.
Consider the bipartite graph on $2|V(G)|$ vertices corresponding to the adjacency matrix of $G$. Since this graph is regular, by Hall's Theorem it has a perfect matching. In terms of the original $G$, this corresponds to a permutation $\sigma$ on $V(G)$ such that $v$ ... | 6 | https://mathoverflow.net/users/405 | 64798 | 40,033 |
https://mathoverflow.net/questions/64795 | 1 | On a tangent to a problem I've been working on, I've run into a combinatorial/partition-theoretic problem that I wondered if anyone had run into before.
Let $N$ be a positive integer, and *ad-hoc*-ly call an (ordered) non-negative partition of $N$ into exactly $N$ parts
$$
N=n\_1+n\_2+\cdots+n\_N
$$
*valid* if
*... | https://mathoverflow.net/users/35575 | Partitions into 0,1, and 2 with a partial sum condition. | This is [A168049](http://oeis.org/A168049) or, with slightly different indexing, the [Motzkin numbers](http://oeis.org/A001006).
The general question of counting nondecreasing sequences which stay below the diagonal is very common in combinatorics and goes by the name Lukasiewicz words. Stanley has a good discussion... | 8 | https://mathoverflow.net/users/297 | 64802 | 40,034 |
https://mathoverflow.net/questions/64812 | 5 | This is a question of terminology. I want to talk about the category whose...
* ...objects are pairs $(G,M)$, where $G$ is a group and $M$ is a $G$-set.
* ...morphisms $(G,M)\rightarrow (G',M')$ are pairs $(f\_G,f\_M)$, where $f\_G:G\rightarrow G'$ is a group homomorphism, and $f\_M:M\rightarrow M'$ is a set map such... | https://mathoverflow.net/users/750 | Is there a 'best' name for a group together with a set it acts on? | How about "actions"? "The category of actions" sounds good to me.
| 13 | https://mathoverflow.net/users/4790 | 64813 | 40,041 |
https://mathoverflow.net/questions/57465 | 73 | The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
1. Is there a binary operation $n\star m$ on the integers
$\mathbb{Z}$ such that both addition $+$ and multiplication
$\cdot$ can be expressed as... | https://mathoverflow.net/users/1946 | Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations? | The answer to Q4 is yes. Let $X$ be any infinite set. Wlog $X= Z\times\mathbb N$, where
there is a bijection $i:X\to Z\times \lbrace0\rbrace $. For $x=(z,n)$ write $x+1$ for $(z,n+1)$. [Typo corrected.]
You are given countably many finitary functions $g\_1, g\_2, \ldots$. We may assume there is a pairing function $x... | 30 | https://mathoverflow.net/users/14915 | 64815 | 40,042 |
https://mathoverflow.net/questions/64817 | 10 | I have computed a generating function for a problem involving a particular series, and would like to know if anyone has any references or a categorisation for it? It's
$$
G(a,z) = \sum\_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}.
$$
It appears to be related to (mock) theta functions, but seems to be simpler.
In particular, I ... | https://mathoverflow.net/users/15056 | Identifying the generating function $ G(a,z) = \sum_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}. $ | It should be noted that using the Jacobi triple product that we have
$$
H(a,z) = a^{-1}\sum\_{n=-\infty}^\infty a^nz^{n(n+1)/2}
= a^{-1}\prod\_{m=1}^\infty (1 - z^m)(1 - z^ma)(1 + z^{m-1}a)
$$
where the main difference is that the indexing shifts and we are doing about twice the sum that you are.
If we try relate the... | 9 | https://mathoverflow.net/users/1703 | 64829 | 40,049 |
https://mathoverflow.net/questions/64823 | 5 | Does the space of homotopy equivalences of $S^1$ deformation retract onto the space of homeomorphisms of $S^1$? If so, does anyone have a reference?
I found that Kneser proved that $Homeo(S^1)$ deformation retracts onto $O(2)$ and $Homeo^+(S^1)$ deformation retracts onto $SO(2)$ (orientation preserving homeos deform... | https://mathoverflow.net/users/15060 | space of homotopy equivalences of $S^1$ | Put $$HE^+\_1(S^1)=\{f\in HE^+(S^1):f(1)=1\}. $$
There is an evident homeomorphism $m:S^1\times HE\_1^+(S^1)\to HE^+(S^1)$ given by $m(z,f)(x)=z f(x)$, and this restricts to give a homeomorphism $S^1\times Homeo\_1^+(S^1)\to Homeo^+(S^1)$.
It will thus suffice to discuss $HE\_1^+(S^1)$ and $Homeo\_1^+(S^1)$.
Ne... | 3 | https://mathoverflow.net/users/10366 | 64836 | 40,053 |
https://mathoverflow.net/questions/64833 | 10 | Is it true that all continuous finite dimensional $p$-adic representations of $Gal(\bar{K}/K)$ are semisimple, where $K$ is a number field, i.e. if $\rho:Gal(\bar{K}/K) \mapsto GL\_n(\bar{\mathbb{Q}}\_p)$ is a continuous representation, where $K$ is a number field (even $\mathbb{Q}$ if you want), is it semisimple?
Re... | https://mathoverflow.net/users/4685 | semisimplicity of p-adic Galois representations | This can't possibly work. Many p-adic Galois representations are not semisimple. For instance, $\mathbb{Z}\_p$ occurs as a quotient of the Galois group of $\mathbb{Q}$ (as the Galois group of the cyclotomic $\mathbb{Z}\_p$-extension) and there are non-semisimple $\mathbb{Q}\_p$-representations of $\mathbb{Z}\_p$, e.g. ... | 19 | https://mathoverflow.net/users/2481 | 64840 | 40,055 |
https://mathoverflow.net/questions/64839 | 4 | Let $X$ be a hyperelliptic curve and let $i:X\to X$ denote the hyperelliptic involution. Once we fix a point $x\_0\in X$ we get the Abel-Jacobi map $AJ:X\to J$ where $J$ denotes the Jacobian variety. Now the Jacobian is also equipped with an involution, namely $x\mapsto x^{-1}$. Is it possible to choose the base point ... | https://mathoverflow.net/users/11395 | Involution on Hyperelliptic curves and their Jacobians | Yes, pick $x\_0$ to be a Weierstrass point, i.e. a fixed point of the hyperelliptic involution.
Let $\sigma$ denote the hyperelliptic involution on $X$. Under the Abel-Jacobi map we have $x \mapsto [x-x\_0]$ and $\sigma(x) \mapsto [\sigma(x) - x\_0]$. Now $[x + \sigma(x) - 2x\_0]$ is the divisor of a function, since... | 7 | https://mathoverflow.net/users/1310 | 64845 | 40,058 |
https://mathoverflow.net/questions/64843 | 6 | I've been reading Hatcher's survey "A short exposition of the Madsen-Weiss theorem". In it, he outlines a nice proof of the "generalized Mumford conjecture", which asserts that the stable cohomology of the mapping class group is the same as the cohomology of $\Omega^{\infty} AG\_{\infty,2}^{+}$. Here $AG\_{n,m}$ is the... | https://mathoverflow.net/users/15068 | Cohomology of the infinite loop space of the affine grassmanian (as in the generalized Mumford conjecture) | Most of this is not special to the case of $AG\_{\infty,2}^+$. For any spectrum $X$, we have a Hurewicz map $h:\pi\_{\ast}(X)\to H\_{\ast}(X)$, which induces a map $h':\mathbb{Q}\otimes\pi\_{\ast}(X)\to\mathbb{Q}\otimes H\_{\ast}(X)$. Standard calculations show that $h'$ this is an isomorphism when $X=S^n$ for some $n$... | 7 | https://mathoverflow.net/users/10366 | 64851 | 40,064 |
https://mathoverflow.net/questions/64810 | 8 | Hi. Does anyone know if it is possible to enumerate the set of labeled/unlabeled graphs (loopless, undirected, only one edge between pairs of nodes) having a given number of nodes, edges and triangles? what about including more information, like number of tetrahedra, etc.? If not, why?
The solution to the unlabeled ... | https://mathoverflow.net/users/15054 | Number of graphs with a given number of nodes, edges and triangles | Polya theory provides us with an algorithm that allows us to compute the number of isomorphism classes of graphs with $n$ vertices and $m$ edges, for given $n$ and $m$. It does not provide a formula in terms of $n$ and $m$ and it does not even provide a generating function. (Well, we can express it in terms of so-calle... | 6 | https://mathoverflow.net/users/1266 | 64858 | 40,067 |
https://mathoverflow.net/questions/64828 | 7 | Hello,
I am wondering if there is a simple asymptotic formula for
$$\sum\_{p\leq x}\frac{\left(\log p\right)^{k}}{p},$$
where $k\geq0$ is some integer. If $k$ is $0,$ by using the Prime Number Theorem we have
$$\sum\_{p\leq x}\frac{1}{p}=\log \log x+b+O\left(e^{-c\sqrt{\log x}}\right).$$
Similarly, the prime... | https://mathoverflow.net/users/12176 | Asymptotic Formula for a Mertens Style Sum | Expanding on Frank's answer: by partial summation, we have that
$$\sum\_{p \leq x} \frac{(\log p)^k}{p} = \frac{(\log x)^k}{x} \pi(x) - \int\_{2}^{x} \pi(t) \left(\frac{kt (\log t)^{k-1} - (\log t)^k}{t^2}\right) dt.$$
Using the fact that $\pi(x) = \mathrm{Li}(x) + E(x)$, where $E(x) = O(e^{-c\sqrt{\log x}})$, we have ... | 6 | https://mathoverflow.net/users/3803 | 64860 | 40,069 |
https://mathoverflow.net/questions/64837 | 3 | Keller and Ochsenius (1995) has a spectral theorem for finite-dimensional symmetric matrices over the field of formal power series with real coefficients $\mathbf{R}((t))$ (they actually have a more general result, but I'm just interested in this one for now), in the sense that every finite symmetric square matrix can ... | https://mathoverflow.net/users/81883 | spectral theorem for infinite-dimensional matrices | In fact, the field considered by Keller and Ochsenius is more complicated; it must have a Krull valuation with a specific kind of value group. They had several papers devoted to the infinite-dimensional case too:
H. Keller and H. Ochsenius, Spectral decompositions of operators on non-Archimedean orthomodular spaces. ... | 3 | https://mathoverflow.net/users/12205 | 64867 | 40,074 |
https://mathoverflow.net/questions/64866 | 28 | This question is about Tate's 1963 paper "Algebraic Cycles and Poles of Zeta Functions". Here he announces a conjecture (now known as "the Tate conjecture") which states that certain classes in the cohomology of a projective variety are always explained by the existence of algebraic cycles. In the case of a variety $X/... | https://mathoverflow.net/users/271 | How does Tate verify his own conjecture for the Fermat hypersurface? | I don't know how Tate did it but here is one way. Let $\zeta$ be such that
$\zeta^{q+1}=-1$ and put $a\_j=(0\colon\cdots\colon1\colon\zeta:\cdots\colon0)$,
$j=0,\ldots,i$ with the $1$ in coordinate $2j$. Then the linear span $L$ of
these points is contained in the Fermat hypersurfaces and gives a subvariety of
middle d... | 25 | https://mathoverflow.net/users/4008 | 64870 | 40,076 |
https://mathoverflow.net/questions/64856 | 4 | I believe a *recursive (partial) functional* $F:\mathbb{N}^\mathbb{N}\to\mathbb{N}$ is ordinarily defined as one for which the "graph" relation $F(\alpha)=n$ is recursively enumerable, which means it can be expressed in the form
$$F(\alpha)=n \iff \exists x.Q(\overline{\alpha}(x), n, x)$$
for some (primitive?) recurs... | https://mathoverflow.net/users/302 | Defining computability for functionals of partial oracles | This sounds like Jaap van Oosten's partial combinatory algebra $\mathcal{B}$, or its effective version, to be precise. You can read about it in John Longley's [survey paper](http://homepages.inf.ed.ac.uk/jrl/Research/notions1.pdf%5D), as mentioned by Ulrik in the comments, or specifically in John's "[Sequentially reali... | 4 | https://mathoverflow.net/users/1176 | 64871 | 40,077 |
https://mathoverflow.net/questions/64874 | 15 | A Riemannian manifold $(M,g)$ is *locally conformally flat* if it is locally conformal to $\mathbb{R}^n$ with the flat metric. I learn that Weyl tensor of a locally conformally flat manifold must vanish. I would like to ask: Is there any example of manifold $M$ such that it cannot be equipped with a metric $g$ with $(M... | https://mathoverflow.net/users/14579 | locally conformally flat manifold | The simplest example is $S^n$, it is locally conformally flat with the standard metric,
and is not flat for obvious reasons.
While flat manifolds are precisely quotients of $\mathbb R^n$ by discreet group of isometries, one should not expect to have a classification of conformally flat manifolds in higher dimensions.... | 20 | https://mathoverflow.net/users/943 | 64880 | 40,080 |
https://mathoverflow.net/questions/64873 | 4 | Hello!
We know that we have an alternative way to define a complex structure on manifold, by means of an integrable almost complex structure. The two points of view are equivalent, this is the content of the Newlander-Nirenberg theorem (which is very difficult to prove).
I think that there is the same kind of notio... | https://mathoverflow.net/users/14806 | Newlander-Nirenberg theorem for general vector bundles | I think I do not exactly understand the question. A complex vector bundle on a manifold is the same as a real vector bundle, together with an endomorphism $J$ with $J^2 =-1$. This is elementary linear algebra and does not deserve to be called an analog of Nirenberg-Newlander theorem.
A different question is whether o... | 12 | https://mathoverflow.net/users/9928 | 64882 | 40,081 |
https://mathoverflow.net/questions/64887 | 6 | Let $f:X\rightarrow Y$ be a birational morphism between smooth varieties and $F$ a torsion free sheaf. Is $f\_{\ast} F$ torsion free? If not, are there conditions on either $F$, $f$, $X$ or $Y$ that could ensure the torsion freeness of $F$? For example if $f$ is surj. and $F$ is the canonical bundle on $X$ a theorem of... | https://mathoverflow.net/users/6949 | Torsion freeness and birational maps | Actually much more is true. In fact there is the following result, whose proof can be found in
[Grothendieck - Dieudonné: EGA 1 (Elements de Géométrie Algebrique), Proposition 8.4.5 page 351].
>
> **Proposition.** Let $X$, $Y$ be two integral schemes and $f \colon X \to Y$ be a dominant morphism. Then for any to... | 8 | https://mathoverflow.net/users/7460 | 64890 | 40,083 |
https://mathoverflow.net/questions/64348 | 6 | Is there an $f\in L^{1}(T)$ whose partial sums of Fourier series $S\_{n}(f)$ satisfies $\|S\_{n}(f)\|\_{L^{1}(T)} \rightarrow \|f\|\_{L^{1}(T)}$ but $S\_{n}(f)$ fails to converge to $f$ in $L^1$-norm ?
| https://mathoverflow.net/users/13244 | On the Existence of Certain Fourier Series | I was hesitating for a while whether to answer or to vote to close and to refer the OP to AoPS, but, since the question has been upvoted, here goes.
Suppose that $f\_k\in L^1$ converges to $f\in L^1$ in the sense of distributions and in measure. Suppose also that $\|f\_k\|\_1\to \|f\|\_1$. Then $f\_k\to f$ in $L^1$. ... | 7 | https://mathoverflow.net/users/1131 | 64897 | 40,088 |
https://mathoverflow.net/questions/64395 | 7 | Let G be a directed graph on N vertices chosen at random, conditional on the requirement that the out-degree of each vertex is 1 and the in-degree of each vertex is either 0 or 2. The "periodic" points of G are those contained in a cycle. What do we know about the statistics of G? For instance, what is the mean number ... | https://mathoverflow.net/users/431 | Dynamics of a random "quadratic" directed graph | Let $G$ be a directed graph on $N$ vertices such that the out-degree of each vertex is 1 and the in-degree is either 0 or $n$. Letting $N=nt$, there are $t$ vertices with in-degree $n$ and $(n-1)t$ vertices with in-degree 0. Assuming the vertices are labeled, the number of such graphs is
$$
\binom{nt}{t}\frac{(nt)!}{(n... | 4 | https://mathoverflow.net/users/752 | 64904 | 40,093 |
https://mathoverflow.net/questions/64896 | 2 | What are the (cont.) endomorphisms resp. automorphisms of the adeles (for a given global field)
1) as a topological abelian group and
2) as a topological ring?
3) What are the endomorphisms and the automorphisms group of the ideles?
4) What is known for the adelic points of an algebraic group?
| https://mathoverflow.net/users/10400 | Endomorphism ring of the adeles and ideles? | 1. There are no nontrivial continuous homomorphisms between factors of different residue characteristic, so any endomorphism/automorphism decomposes into a collection of endomorphisms of factors, with a global condition that $(1,1,\cdots)$ has bounded denominators. Each factor over a rational prime $p$ (including infin... | 2 | https://mathoverflow.net/users/121 | 64908 | 40,096 |
https://mathoverflow.net/questions/64709 | 6 | Let $P$ be a polynomial of fixed degree $d$ with integer coefficients of absolute values at most $n$. Assume that $P(\cos 2\pi/n)$ is no zero. Is there a lower bound for $|P(\cos 2\pi/n)|$ ? For instance, is this at least $n^{-C(d) }$ where $C(d)$ is some constant depending on $d$ ?
| https://mathoverflow.net/users/15028 | Diophantine approximation | It seems one can get something by just looking at the Taylor expansion of $\cos$. Let me consider $\cos 1/n$ for simplicity. Plugging
$$\cos 1/n = 1- (1/n)^2/2! + (1/n)^4/ 4! -.... $$
into $P= a\_d x^d + a\_{d-1} x^{d-1}+ ...+ a\_1 x + a\_0 $ one has
$$(a\_d+ ...+ a\_0) - \frac{(1/n)^2}{2!} (da\_d +...+ a\_1) -... | 0 | https://mathoverflow.net/users/15028 | 64915 | 40,101 |
https://mathoverflow.net/questions/64925 | 7 | Suppose $G$ is a group and $V$ an irreducible representation of $G$. One has that $V\otimes V\cong \Lambda^2(V)\oplus Sym^2(V)$. It is well-known that if the trivial representation appears as a subrepresentation of $\Lambda^2(V)$ then $V$ is of quaternionic type; while if the trivial representation appears as a subrepr... | https://mathoverflow.net/users/12301 | Occurence of trivial representation in a tensor square. | The trivial representation appears in $\wedge^2 V$ if and only if the representation $V^{\ast}$ has a $G$-invariant alternating bilinear form (because $\wedge^2 V\cong\wedge^2\left(\left(V^{\ast}\right)^{\ast}\right)$ is isomorphic to the $G$-module of all alternating bilinear forms on $V^{\ast}$, and $G$-invariant for... | 7 | https://mathoverflow.net/users/2530 | 64926 | 40,107 |
https://mathoverflow.net/questions/64909 | 3 | Let $X$ be a smooth geometrically irreducible $k$-variety over a number field $k$.
I do not assume that $X$ has a $k$-point.
Is it true that $X$ has $k\_v$-points for almost all places $v$ of $k$?
| https://mathoverflow.net/users/4149 | Rational points over completions of a number field | Without loss of generality $X$ is affine, so embed it in projective space and apply the Bertini Theorem to conclude that $X$ contains a smooth, geometrically integral affine curve $C^{\circ}$ missing $N$ points from its projective completion $C$. Such a guy will remain smooth modulo $v$ for almost all places of $v$ -- ... | 13 | https://mathoverflow.net/users/1149 | 64929 | 40,109 |
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