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/140306 | 7 | I am trying to do some experimentation with the values of Thompson series, but I have having a hard time finding a table that has these Thompson series with as many terms as I'd like. The tables I've seen only give me the required head character values up to the $q^{10}$ term.
Can someone point me to a resource that ... | https://mathoverflow.net/users/38058 | Computing Thompson series for the monster group | From MathSciNet:
MR1037906 (90m:11065) Reviewed
McKay, John(3-CONC); Strauss, Hubertus(3-CONC)
The q-series of monstrous moonshine and the decomposition of the head characters.
Comm. Algebra 18 (1990), no. 1, 253–278.
11F22 (20C15 20D08 33A99)
The authors tabulate the first fifty coefficients of the q-expansion... | 11 | https://mathoverflow.net/users/10475 | 140308 | 76,693 |
https://mathoverflow.net/questions/140310 | 2 | A possible formulation of the Bestvina-Feighn theorem is as follows (taken from [here](http://392c.wordpress.com/2009/04/08/29-doubles-of-free-groups/)):
**Combination Theorem (Bestvina & Feighn):** If $H$ is a malnormal subgroup of hyperbolic groups $G\_1, G\_2$, then $G\_1\ast\_H G\_2$ is hyperbolic.
I was wonder... | https://mathoverflow.net/users/29775 | Amalgmated free product of hyperbolic groups with one malnormal and one virtual factor is hyperbolic? | As Misha says in comments, as long as H is also quasiconvex in $G\_1$ then you will be able to apply the combination theorem. One can deduce this directly from the 'usual' statement that an amalgam of two hyperbolic groups along malnormal, quasiconvex subgroups is itself hyperbolic, as follows.
If $H$ is finite-index... | 4 | https://mathoverflow.net/users/1463 | 140314 | 76,696 |
https://mathoverflow.net/questions/139738 | 7 | In the book, "Pi and the AGM" by Borwein and Borwein, it is mentioned that Gauss computed the following integral to the eleventh decimal palce.
$\int\_0^1 \frac{1}{\sqrt{1-x^4}}dx$
How did he do it? Personnally, I looked at a Taylor expansion of
$\frac{1}{\sqrt{1-x}}$
Where I substituted $t^4$ for $x$, and inte... | https://mathoverflow.net/users/38798 | Numerically computing $\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$ | A good place to look is pages 405 and 413 of the *Nachlass* section of Gauss's *Werke III*, which can be found online through Google Books. On page 405, he gives the following formula for "$\text{arc sin lemn }x$":
$$\text{arc sin lemn }x= x+{1\over2}\cdot{1\over5}x^5 + {1\cdot3\over2\cdot4}{1\over9}x^9+{1\cdot3\cdot... | 8 | https://mathoverflow.net/users/15837 | 140317 | 76,699 |
https://mathoverflow.net/questions/139643 | 6 | Assume that ${\mathbf H}$ is a $N \times M$ matrix. The following parameter is called orthogonality deficiency and describes how much orthogonal the columns of ${\mathbf H}$ are.
$$ od({\mathbf H}) = 1 - \frac{\det({{\mathbf H}^H{\mathbf H})}}{\Pi\_{n=1}^M\|{{\mathbf h}\_n}\|^2}$$
where ${\mathbf h}\_n$ is the $n$th c... | https://mathoverflow.net/users/38730 | Taylor expansion of a function of a matrix | If $e$ is a small number, then $od(H)\approx od(A)+e(od)'\_A(B)$. $od(A)=1-\dfrac{u(A)}{v(A)}$
where $u(A)=\det(A^\*A),v(A)=\Pi\_i||Ae\_i||^2$ and $(e\_i)\_i$ is the canonical basis.
$(od)'\_A=-\dfrac{1}{v(A)}u'\_A+\dfrac{u(A)}{v^2(A)}v'\_A$.
$u'\_A(K)=trace((A^\*K+K^\*A)adjoint(A^\*A))$.
$v'\_A(K)=\sum\_i((e\_i^\*K^\... | 2 | https://mathoverflow.net/users/9091 | 140326 | 76,700 |
https://mathoverflow.net/questions/140293 | 0 | This question is related to this one [A question about the inverse theorem function in $\mathbb{R}^n$](https://mathoverflow.net/questions/140207/a-question-about-the-inverse-theorem-function-in-mathbbrn). In the response I received, I was told that the issue was related to the singularities, as I do not have a strong b... | https://mathoverflow.net/users/24060 | Application of the inverse theorem function on singular points | Near any point where $V$ has a zero of order 1, i.e. a point where precisely two of the $t\_i$ are equal, the map looks like a fold singularity, so doesn't have any such neighborhood. It looks like $(x,y,\dots) \mapsto (x^2,y,\dots)$. I think you can find everything you need in [this nice introduction](http://www.sissa... | 2 | https://mathoverflow.net/users/13268 | 140328 | 76,701 |
https://mathoverflow.net/questions/140327 | 27 | Arnold, in his paper
The underestimated Poincaré, in Russian Math. Surveys 61 (2006), no. 1, 1–18
wrote the following:
>
> ``...Puiseux series, the theory which Newton, hundreds of years before Puiseaux,
> considered as his main contribution to mathematics (and which he encoded as a second,
> longer anagram, des... | https://mathoverflow.net/users/25510 | Arnold on Newton's anagram | Newton's anagram on his method to solve differential equations is contained in his letter to Leibniz dated October 24, 1676, as described [here](http://www.maths.tcd.ie/pub/HistMath/People/Newton/RouseBall/RB_Newton.html)
>
> At the end of his letter Newton alludes to the solution of the
> "inverse problem of tan... | 30 | https://mathoverflow.net/users/11260 | 140332 | 76,702 |
https://mathoverflow.net/questions/140319 | 3 | Is it known when a Nakajima quiver variety happens to be a local complete intersection?
[For simplicity consider an affine quiver variety, i.e. the categorical quotient of the zero set of the moment map on the space of representations of a doubled quiver]
| https://mathoverflow.net/users/24483 | are quiver varieties local complete intersections? | By Proposition 1.3 of Beauville's original [article](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEkQFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.54.9491%26rep%3Drep1%26type%3Dps&ei=f0AZUp2wJOT54AOBxYDoAw&usg=AFQjCNEFQNxgm0ekulgib2n6CGTuvEq_sA&sig2=0ywjNlBzWaFe... | 6 | https://mathoverflow.net/users/66 | 140335 | 76,703 |
https://mathoverflow.net/questions/140338 | 3 | I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical imaging/imaging problems in particular. I searched it online, but it was not so productive for me.
I was also wondering wheth... | https://mathoverflow.net/users/35936 | Request for some references exploring the connections of Riemann surfaces with medical imaging | Here are some papers--accessible to beginners-- relating circle packings (which themselves are related to triangulations of Riemann surfaces) and image processing:
MR2492509 Williams, G. Brock: Circle packings, quasiconformal mappings and applications. Quasiconformal mappings and their applications, 327–346, Narosa, ... | 3 | https://mathoverflow.net/users/14493 | 140340 | 76,704 |
https://mathoverflow.net/questions/138583 | 3 | Let $\cal C$ be the smallest class of finitely generated discrete countable groups such that
1. $\cal C$ contains all infinite cyclic groups,
2. if $G$ is *any* finitely generated countable discrete group and $H\in \cal C$ then $G\ast H \in \cal C$,
3. If $G$ is any countable discrete group and $H$ is a finite index ... | https://mathoverflow.net/users/2631 | Is there a better description of this class of discrete groups? | $\cal C$ is the class of finite extensions of finitely generated groups having an infinite cyclic free factor.
(as in Mark's comment concerning another version of the question).
To prove this it suffice to note that the class of such finite extensions is closed with respect to finite-index subgroup. Indeed,
a fin... | 4 | https://mathoverflow.net/users/24165 | 140345 | 76,707 |
https://mathoverflow.net/questions/140348 | 2 | For a metric space $E$, let $\mathcal{H}(E)$ be the metric space consisting of the set of nonempty compact subsets of $E$ and the Hausdorff metric. Consider the following two statements.
1. Let $X$ and $Z$ be topological spaces and let $Y$ be a compact metric space. Let $f : X \times Y \rightarrow Z$ be a continuous ... | https://mathoverflow.net/users/10271 | Continuity with Hausdorff metric | 1. Is false. Consider $Y = Z = [0,1]$, $X = [0,1] \cup \{2\}$ and
$$f(x,y) = \begin{cases} y & \mbox{if }x = 2 \\ x & \mbox{otherwise}\end{cases}$$ Then $g(y,z) = \{x: f(x,y) = z\}$ is either $\{2,z\}$ if $y=z$ or $\{z\}$ if not, and this is not continuous.
2. is quite easy, because $|\min(A) - \min(B)| \le \mbox{dist... | 3 | https://mathoverflow.net/users/13650 | 140349 | 76,709 |
https://mathoverflow.net/questions/139832 | 4 | An ideal $J$ of a ring $A$ is a heredity ideal of $A$ if:
* $J^2=J$;
* $J$ is a projective $A$-module;
* $J (\operatorname{Rad}A)J=0$.
A (unitary) semiprimary ring $A$ is said to be quasihereditary if there exists a chain of ideals of $A$,
$$0=J\_0 \subseteq \cdots \subseteq J\_i \subseteq \cdots J\_n =A,$$
such th... | https://mathoverflow.net/users/36805 | Are the heredity ideals in an heredity chain always finitely generated? | Yes, it is possible. (So the answer to the question in the title is no.)
Consider the ring
$$A = \begin{bmatrix} \mathbb Q & \mathbb R \\ 0 & \mathbb Q \end{bmatrix}.$$
It has primitive idempotents $e=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ and $f=\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}.$
Now $J=AeA$ is... | 4 | https://mathoverflow.net/users/18756 | 140356 | 76,712 |
https://mathoverflow.net/questions/140358 | 6 | Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are homeomorphic. But I wonder how independent $X$ and $Y$ could be ? For example is there any forced relation between their cardin... | https://mathoverflow.net/users/nan | Isomorphic rings of functions | Let $X,Y$ be arbitrary sets (of arbitrary cardinals) armed with topologies $\tau\_1 = \lbrace \emptyset, X\rbrace$ and $\tau\_2 = \lbrace \emptyset, Y\rbrace$. Then it is clear that $C(X) \cong \Bbb{R} \cong C(Y)$. So there is no forced relation between the cardinal numbers.
| 9 | https://mathoverflow.net/users/nan | 140360 | 76,713 |
https://mathoverflow.net/questions/127773 | 9 | Suppose $G = (V,E)$ is a directed graph.
For sets $A$ and $B$ of vertices of $G$,
let $d(A,B) = |(A \times B) \cap E| / (|A||B|)$ denote the edge density between $A$ and $B$,
and say that the pair $A,B$ is $\epsilon$-*regular* if
$$ |d(X,Y) - d(A,B)| \lt \epsilon $$
whenever $X \subseteq A, Y \subseteq B$, $X$ contains... | https://mathoverflow.net/users/7252 | In Szemerédi's Regularity Lemma, how many blocks are in the partition? | In addition to Gowers' breakthrough paper, a recent paper addressing the bounds in Szemerédi's regularity lemma, and its weak and strong variants is:
D. Conlon and J. Fox, Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012), 1191-1256.
It is shown that the bound on the number of parts for ... | 7 | https://mathoverflow.net/users/39117 | 140383 | 76,723 |
https://mathoverflow.net/questions/138575 | 15 | I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the Euclidean space $\mathbb{R}^n$.
I was told that for general symmetric spaces there is a Harish-Chandra type theorem saying... | https://mathoverflow.net/users/16183 | Invariant differential operators on real Grassmannians | The answer to my original question of description of the algebra of $O(n)$-invariant operators on real Grassmannians is explicitly contained in Proposition 3.2 of the paper by
Fulton B. Gonzalez, Tomoyuki Kakehi; "Pfaffian systems and Radon transforms on affine Grassmann manifolds", Mathematische Annalen 326 (2003).
| 1 | https://mathoverflow.net/users/16183 | 140388 | 76,726 |
https://mathoverflow.net/questions/140373 | 9 | Gentzen's Hauptsatz in first order logic includes an algorithm taking any proof in the sequent calculus with cut rule, and delivering a proof without cut rule (and with the subformula property). So far as I know cut elimination in the simple theory of types (STT) has no such algorithm. Rather, one proves non-constructi... | https://mathoverflow.net/users/38783 | Cut elimination algorithms | As Noah pointed out, if it's provable, there must be an algorithm for cut-elimination for STT because the underlying statement is $\Pi\_2$ (for any deduction in STT, there is a corresponding deduction without cut). Moreover, if you can prove cut-elimination for STT in some theory T for which you do have an algorithm fo... | 4 | https://mathoverflow.net/users/8991 | 140392 | 76,728 |
https://mathoverflow.net/questions/140374 | 12 | In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum\_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e.
My question is: what is, given $\epsilon>0,$ the estimation of $\mu\left(x:\left|\frac{1}{N}\sum\_{n=0}^{N-1}f(T^n x)-\int f ... | https://mathoverflow.net/users/39115 | Birkhoff ergodic theorem and the measure of the bad points | The key words here are "large deviations"; large deviations theory addresses exactly this question. The answer depends quite a bit on the specific measure and system in question, but roughly speaking one may say the following: if the system displays a sufficient amount of hyperbolic behaviour (for example, an Axiom A s... | 17 | https://mathoverflow.net/users/5701 | 140395 | 76,729 |
https://mathoverflow.net/questions/140386 | 4 | Let $M$ be a Riemannian manifold with totally geodesic boundary $\partial M$. We let $\check{M}$ be its double, i.e. the disjoint union of $M$ with itself under identification of corresponding boundary points. It is then well known that $\check{M}$ is a smooth manifold. Can you tell me whether the metric is still smoot... | https://mathoverflow.net/users/39122 | Regularity of metric of the double of a Riemannian manifold | You should take a look at some of the answers to previous questions here that address this point, especially [this one](https://mathoverflow.net/questions/67809/the-double-of-a-smooth-manifold-with-boundary) and [this one](https://mathoverflow.net/questions/82270/reference-request-gluing-manifolds-along-pieces-of-bound... | 5 | https://mathoverflow.net/users/15743 | 140397 | 76,731 |
https://mathoverflow.net/questions/140401 | 2 | Pardon my ignorance of this topic.
>
> **Q1**.
> In which dimensions $d$ is it the case that, for every natural number $n$,
> there exists a sphere having exactly $n$ lattice points on it $(d{-}1)$-dimensional surface?
>
>
>
[Schinzel's Theorem](http://mathworld.wolfram.com/SchinzelsTheorem.html) establishes... | https://mathoverflow.net/users/6094 | Three questions concerning lattice points on sphere surfaces | Collecting my comments into an answer:
It appears that Kulikowski's paper proved the result for all dimensions. I haven't seen the Kulikowski paper, but a short proof is given at this [site](http://www.cut-the-knot.org/arithmetic/algebra/Kulikowski.shtml).
Question 2 is well-answered by Lev Borisov elsewhere on t... | 3 | https://mathoverflow.net/users/3684 | 140404 | 76,733 |
https://mathoverflow.net/questions/139140 | 63 | The original post is below. Question 1 was solved in the negative by David Speyer, and the title has now been changed to reflect Question 2, which turned out to be the more difficult one. A bounty of 100 is offered for a complete solution.
**Original post.** It follows from the prime number theorem and the periodicit... | https://mathoverflow.net/users/26522 | Are there infinitely many integer-valued polynomials dominated by $1.9^n$ on all of $\mathbb{N}$? | Question 2:
The constant $A$ can be brought down to $\sqrt 3$, and probably
a bit below that but not all the way down to $1+\epsilon$.
Instead of the polynomial $f(n) = {n \choose m}$, use a
finite difference of such polynomials,
$$
f(n) = \sum\_{i=0}^m (-1)^i {m \choose i} {n \choose m+i}.
$$
This is the $x^n$ coef... | 47 | https://mathoverflow.net/users/14830 | 140411 | 76,736 |
https://mathoverflow.net/questions/140272 | 9 | The question is really simple:
Given
$$
f, g\in C^\alpha\_c(\mathcal{R}^d)
$$
is
$$
f\*g\in C^d\_c?
$$
I came up with a formal argument using the decay of the Fourier transform of continuous functions, but it is really formal and I would appreciate a reference.
What I have though so far, which has a few holes is t... | https://mathoverflow.net/users/39062 | How differentiable is the convolution of two continuous functions? | $C^\alpha\_c\*C^\beta\_c\subset C^{\alpha+\beta}\_c$ but not much better than that. If you want it through Fourier analysis, the shortest route to go is to periodize and to use the Bernstein description of periodic $C^\alpha$ with non-integer $\alpha$ as the set of continuous functions $f$ such that for every $n$, ther... | 7 | https://mathoverflow.net/users/1131 | 140412 | 76,737 |
https://mathoverflow.net/questions/140418 | 3 | Can someone give me a short proof of the identity,
$$\sum\_{n=1}^\infty\frac{q^nx^{n^2}}{1-qx^{n}}+\sum\_{n=1}^\infty\frac{q^nx^{n(n+1)}}{1-x^n}=\sum\_{n=1}^\infty\frac{q^nx^n}{1-x^n}$$
| https://mathoverflow.net/users/38626 | Lambert series identity | It's just about using geometric series a lot. Indeed, we have
$\sum\limits\_{n=1}^\infty\frac{q^n x^{n^2}}{1-qx^n}=\sum\limits\_{n=1}^\infty\sum\limits\_{k=0}^\infty q^{n+k}x^{n(n+k)}=\sum\limits\_{m=1}^\infty\sum\limits\_{d\ge m}q^dx^{md}$
and
$\sum\limits\_{n=1}^\infty\frac{q^n x^{n(n+1)}}{1-x^n}=\sum\limits\_{... | 5 | https://mathoverflow.net/users/1306 | 140421 | 76,739 |
https://mathoverflow.net/questions/140389 | 5 | Let $p$ be a prime number $\geq 3$. Let $V$ be a representation of $Gal(\bar{\mathbb{Q}}\_p/ \mathbb{Q}\_p)$ with coefficients in $\mathbb{F}\_p$. Assume $V$ is a non-split extension of two characters $\delta\_1$ and $\delta\_2$.
Let $D$ be the $(\varphi, \Gamma)$-module associated to $V$. It is a non-split extension... | https://mathoverflow.net/users/38010 | $(\varphi, \Gamma)$-module of dimension 2 modulo $p$ | The answer is "no" in general. If $\delta\_2=1$ and $\delta\_1$ is the mod $p$ cyclotomic character, then there are both peu and très ramifiées extensions. Let $res(g)$ denote the coefficient of $1/X$ in $g$. Theorem: in your notation, the extension is peu ramifiée iff $res(g)=0$. See for instance proposition 3.7.5 of ... | 4 | https://mathoverflow.net/users/5743 | 140429 | 76,742 |
https://mathoverflow.net/questions/140352 | 17 | Suppose $\mu$ and $\nu$ are two probability measures on $[0,1]$. Let their $n$-th moments be denoted by $\mu\_n$ and $\nu\_n$, respectively, for $n \in \mathbb{N}$.
If we know that $\mu\_n=\nu\_n$ for infinitely many $n$, can we conclude that $\mu=\nu$?
One way to resolve this would be to see if the span of $\{ x^n... | https://mathoverflow.net/users/37273 | A moment problem on $[0,1]$ in which infinitely many moments are equal | We actually have that $\mu=\nu$ is guaranteed if and only if $\sum\_{n\in S}\frac 1n$ is divergent. It's a condition which translates the fact that the set of indexed $k$ such that $\mu\_k=\nu\_k$ has to be large enough.
Recall [Müntz-Szász](http://en.wikipedia.org/wiki/M%C3%BCntz%E2%80%93Sz%C3%A1sz_theorem) theorem,... | 15 | https://mathoverflow.net/users/17118 | 140432 | 76,743 |
https://mathoverflow.net/questions/140430 | 1 | Let $(A^\mathbb{N}, \mathcal{B}(A^\mathbb{N}), \mu)$ be a measure space, where $A^\mathbb{N}$ is a set of one-sided sequences over a finite alphabet $A \subset \mathbb{N}$, $\mathcal{B}(A^\mathbb{N})$ is the Borel sigma-algebra generated by cylinder sets and $\mu$ is a measure with full support (say, Bernoulli or Marko... | https://mathoverflow.net/users/39149 | A set of positive-measure not being a countable union of cylinder sets and zero-measure sets? | Let $\mu$ be a non-atomic Borel probability measure on a Polish space $X$, and let $\delta>0$. Then there exists a closed set $K \subset X$ such that $\mu(K)>1-\delta$ and $K$ has empty interior.
To see this we argue as follows. Let $(x\_n)\_{n=1}^\infty$ be a sequence of distinct points which is dense in $X$. For ea... | 4 | https://mathoverflow.net/users/1840 | 140433 | 76,744 |
https://mathoverflow.net/questions/140264 | 17 | It is well known that the infinite sum:
$$\displaystyle \zeta(s) = \sum\_{n=1}^\infty \frac{1}{n^s}$$
only converges for $\Re(s)>1$.
The Dirichlet 'alternating' sum:
$$\displaystyle \zeta(s) = \frac{1}{1-2^{1-s}}\sum\_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}$$
allows for analytic continuation towards $\Re(s)>0$, ... | https://mathoverflow.net/users/12489 | Does this infinite sum provide a new analytic continuation for $\zeta(s)$? | This can be construed as an application of Euler-Maclaurin summation, as carried out to an arbitrary number of stages in Appendix B of Montgomery-Vaughan's "Multiplicative Number Theory I: Classical Theory".
The general idea is to subtract the integral corresponding to a sum, breaking the integral into integrals over... | 16 | https://mathoverflow.net/users/15629 | 140449 | 76,748 |
https://mathoverflow.net/questions/140359 | -4 | Let $G$ be a finite group. Several recent papers (see e.g. <http://www.jstor.org/discover/10.2307/2695441>) deal with the following notion: $G$ is called a *group with perfect order subsets* or briefly, a *POS-group* if the number of elements of any possible order in $G$ is a divisor of $|G|$. Note that the symmetric g... | https://mathoverflow.net/users/17565 | A question on the number of subgroups of symmetric groups | The answer is yes for $n=1,2,3$ only.
The group $S\_n$ has an elementary abelian subgroup $H$ of order $2^{\lfloor n/2 \rfloor}$ generated by the transpositions $(1,2), (3,4), \ldots,$. You can check that $H$ has at least $2^{\lfloor n/4 \rfloor \lfloor (n+2)/4 \rfloor}$ subgroups of order $2^{\lfloor n/4 \rfloor}$.
... | 5 | https://mathoverflow.net/users/35840 | 140458 | 76,753 |
https://mathoverflow.net/questions/140459 | 57 | There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current proofs of FLT is their use of the modularity thesis which is just the opposite: arcane, and richly connected to a lot of resul... | https://mathoverflow.net/users/38783 | Has Fermat's Last Theorem per se been used? | Corollary 3.17 in [this paper of Stefan Keil](http://arxiv.org/abs/1206.1822) uses FLT for exponent 7 to show that if $E/\mathbb{Q}$ is an elliptic curve with a rational 7-torsion point $P$, and $E\rightarrow E'$ is the 7-isogeny with kernel $\langle P\rangle$, then $E'(\mathbb{Q})[7]=0$. There are of course lots of wa... | 45 | https://mathoverflow.net/users/35416 | 140470 | 76,760 |
https://mathoverflow.net/questions/140455 | 0 | I am not a mathematician, I am a programmer. Sorry, if formulation of the problem is inexact.
I want to calculate the probability of winning for a selected tic-tac-toe player.
I have a directed graph of the game, where the vertices of the graph are game positions, directed edges are a moves from one player to another... | https://mathoverflow.net/users/39155 | Calculate the probability of winning for a selected tic-tac-toe player | The formulation of the question is very unclear, so I'll make some guesses about what was meant. First, I assume that all vertices have the same sort of data available, namely how many winning positions (not how many sequences of moves leading to those positions) there are for one or the other player after any number o... | 1 | https://mathoverflow.net/users/6794 | 140471 | 76,761 |
https://mathoverflow.net/questions/140280 | 1 | This is a continuation of this [question](https://mathoverflow.net/questions/138025/centralizer%20of%20the%20order%202%5Ek%20cyclic%20permutation%20matrix%20over%20F_2), where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}\_2$ of the $n$-cycle. We needed to know the expl... | https://mathoverflow.net/users/11100 | centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2) | I don't know if this is in the literature, but I claim that the structure of the unit group of $\mathbb{F}\_q C\_n$ in general can be derived using similar ideas as in the proof of Bass' Proposition XI.5.7. and some standard techniques:
Bass computes the structure of $(\mathbb{F}\_p C\_{p^k} )^\* = \mathbb{F}\_p^\* ... | 2 | https://mathoverflow.net/users/10266 | 140474 | 76,763 |
https://mathoverflow.net/questions/140409 | 3 | On pg. 133 of *Heat Kernels and Spectral Theory*, Davies is studying the heat kernel $K(x,y,t)$ of the operator $H = -\Delta + |x|^{\alpha}$ for $\alpha > 0$. He wishes to prove a lower bound, and writes, "If $H\_{B}$ is the operator obtained from $H$ by imposing Dirichlet boundary conditions on the surface of the ball... | https://mathoverflow.net/users/12968 | Is there a Feynman-Kac formula applicable to Dirichlet problems for Schrödinger operators? | Yes, if $H = -\Delta + V$ on a smooth domain $D$ with Dirichlet boundary conditions, then the solution to $\partial\_t u = - H u$ with initial condition $u\_0$ is given by
$$
u(t,x) = E\_x \Bigl[ \exp\Bigl(-\int\_0^t V(X\_s)\,ds\Bigr) u\_0(X\_s) 1\_{t < \tau}\Bigr]\;,
$$
where $X$ is a Brownian motion started at $x$ an... | 4 | https://mathoverflow.net/users/38566 | 140478 | 76,766 |
https://mathoverflow.net/questions/117486 | 5 | Is there an infinite family $\lbrace R\_\alpha\rbrace\_\alpha $ of rings (with identity $1\neq 0$) such that
their direct product is a hereditary ring ?
I think the answer must be negative but i have no proof or counterexample yet.
| https://mathoverflow.net/users/nan | Direct product of rings | As @Jeremy Rickard has mentioned, the impossibility was shown by Osofsky in 1968. But it is interesting that there is another article dating back to 1968 (again!) that proves the same thing. See **Cateforis, Sandomerski, The singular submodule splits off, J. Algebra, 10 (1968), 149-165, Theorem 4.1.** This article was ... | 4 | https://mathoverflow.net/users/nan | 140482 | 76,767 |
https://mathoverflow.net/questions/140438 | 22 | We say that two metrics are affinely equivalent if their Levi-Civita connections coincide. Is it possible that an Einstein (=Ricci tensor is proporional to the metric) is affinely equivalent to a metric which is not Einstein?
Of course, since affinely equivalent metrics have the same Ricci tensor,
the question is e... | https://mathoverflow.net/users/14515 | Can an Einstein metric have the same Levi-Civita connection with a non-Einstein one? | There are trivial examples that arise just by taking products of irreducible Einstein metrics with different Einstein constants. Whether an irreducible example exists is a much harder question. I do not know the answer to that, but I can think about it. (*However, see below, where I answer this question.*)
Of course,... | 22 | https://mathoverflow.net/users/13972 | 140483 | 76,768 |
https://mathoverflow.net/questions/140469 | 4 | Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B\_{ij}$ such that vertices $x\_i$ and $x\_j$ are antipodal on this ball. Fix a point $x\_0$ in the simplex. The question: is $x\_0$ in at least $n$ balls?
**Some notes.** The question cannot be strengthened by claiming, for example, that... | https://mathoverflow.net/users/38961 | n-simplex in an intersection of n balls | Yes, because a connected graph ($i\sim j$ iff $\angle x\_ix\_0x\_j$ is more than $\frac\pi 2$) with $n+1$ vertices has at least $n$ edges.
The graph is connected because (assuming WLOG that $x\_0=0$ is strictly inside the simplex) there exists a linear combination $\sum\_j y\_j=0$ where $y\_j=c\_jx\_j$, $c\_j>0$. If th... | 4 | https://mathoverflow.net/users/1131 | 140493 | 76,772 |
https://mathoverflow.net/questions/140481 | 1 | The paper, "The Multinomial-Poisson Transformation" by S. Baker (see <http://www.math.ntnu.no/inla/r-inla.org/papers/multinomial-poisson.pdf>) presents "likelihood kernels" for multinomial variables, and then transforms these kernels to Poisson.
What is a likelihood kernel and how does it relate to the multinomial or... | https://mathoverflow.net/users/39175 | What is a likelihood kernel? | I'm about 90% sure that in this case it means the equivalence class to which the likelihood function belongs, where two functions are equivalent precisely if either is a positive scalar multiple of the other. Notice that
$$
\Pr(Y\_1=y\_1\ \&\ \cdots\ \&\ Y\_J=y\_j) = \frac{(y\_1+\cdots+y\_J)!}{y\_1!\cdots y\_J!}\cdot \... | 1 | https://mathoverflow.net/users/6316 | 140503 | 76,777 |
https://mathoverflow.net/questions/140504 | 2 | Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how the preimage of the generator of $H^n(M)$ looks like.
>
> Is there a general description of the complex bundle $E \t... | https://mathoverflow.net/users/13356 | Preimage of $1 \in H^n(M^n)$ under Chern character | Suppose $M$ is a 2n dimensional manifold and $F$ is a rank $n$ complex vector bundle on $M$ with $c\_n(F)=k \in H^{2n}(M)$. Then
$$ \sum\_{i=0}^n (-1)^i[\Lambda^iF^\*] $$
is an element in $K(M)$ which is the preimage of $k$ under the chern character map. Assuming that $k\neq 0$, this will be the preimage of a gene... | 4 | https://mathoverflow.net/users/9617 | 140519 | 76,784 |
https://mathoverflow.net/questions/140522 | 1 | Let $W$ be a one-dimensional standard Brownian motion and denote $$X\_t=-\mu t + \sigma W\_t, \quad t\ge 0,$$ where $\mu$ and $\sigma$ are positive constants. For $b<0$ denote the first passage time of level $b$ by $\tau$: $$ \tau:=\inf\{t\ge 0: X\_t=b\}.$$ My question is: how can one find $${\mathbb E}\left[\int\_0^\t... | https://mathoverflow.net/users/34483 | On the expectation of a path integral involving Brownian motion up to a random time | Define $u(x)=E^x\int\_0^\tau X\_s ds$, then $u$ satisfies $\sigma^2 u\_{xx}/2-\mu u\_x=-x$ with boundary condition $u(b)=0$ and $u(\infty)=\infty$. (This is missing a boundary condition, but a good way to discover the extra boundary condition at $b$ is to solve first in a strip and then take the width of the strip to i... | 5 | https://mathoverflow.net/users/35520 | 140530 | 76,786 |
https://mathoverflow.net/questions/140520 | 7 | The simplest example of Koszul duality (see introduction of [Beilinson, Ginzburg, and Soergel - Koszul Duality Patterns in Representation Theory](https://doi.org/10.1090/S0894-0347-96-00192-0)) is as follows.
Let $V = \mathbb{C}x$ be a $1$ dimensional vector space. Then the exterior algebra is $A=\mathbb{C}[x]/(x^2)$... | https://mathoverflow.net/users/2623 | Koszul (exterior/symmetric) duality for a 1-dim vector space | "mod" has to mean the category of finitely generated *graded* modules, if not, there is no such equivalence.
Up to graded shifts (and isomorphism), the category $A\text-\mathsf{mod}$ in your example only has two indecomposable objects: the trivial module $\mathbb C$ and the length two projective-injective module $A$.... | 11 | https://mathoverflow.net/users/18756 | 140540 | 76,792 |
https://mathoverflow.net/questions/140561 | 1 | I am revising a paper where one of the operations performed on a undirected graph with no loops, is to take each vertex, and split it into two vertices, and take each edge and replace it with 4 edges: e.g. vertices a and b with edge (a,b) become vertices -a, +a, -b, +b, with edges (-a,-b), (-a,+b), (+a,-b), (+a,+b). A ... | https://mathoverflow.net/users/12911 | Is there a name for this operation on graphs - e.g. duplication, Kronecker product of graphs? | Its the lexicographic product of the empty graph on two vertices by your original graph. (The terminology for lexicographic product is a bit confused - it's not commutative and there are two schools of thought about which graph should go first.)
| 3 | https://mathoverflow.net/users/1266 | 140565 | 76,803 |
https://mathoverflow.net/questions/140568 | 6 | Let $S:=P\Omega\_{2n}^+(q)$ with $n$ even and $q$ odd prime power be the simple orthogonal group. Then the Schur multiplier of $S$ is the Klein four-group $Z\_2\times Z\_2$. Therefore $S$ has three double covers.
Is there any relation between these three double covers? Are they isomorphic? When $n=4$ I know that the... | https://mathoverflow.net/users/32259 | Double covers of the orthogonal groups | Sorry for editing this answer multiple times. However, as I managed to get the answer wrong I feel obliged to improve this answer and provide a few more details. I've broken this up into several parts, so you can read as much as you care about.
---
**My original (incorrect) answer:**
The three double covers of ... | 7 | https://mathoverflow.net/users/22846 | 140569 | 76,805 |
https://mathoverflow.net/questions/140579 | 2 | How could I prove that
$$\sum \_{m=v}^n \left(\left(\prod \_{k=v}^{m-1} \frac{k^2}{m^2-k^2}\right)\left(\prod \_{k=m+1}^n \frac{k^2}{k^2-m^2}\right)(-1)^{m-v}\right)=1$$
or, simplified,
$$\sum \_{m=v}^n \prod \_{k=v, k \neq m}^{n} \frac{k^2}{k^2-m^2}=1$$
for any positive integers $v$ and $n$, $v \leq n$? I feel this ... | https://mathoverflow.net/users/39227 | An identity involving sum of probably binomial coefficients | Consider the contour integral of
$$
\frac{1}{z} \prod\_{k=v}^{n} \frac{k^2}{k^2-z^2}
$$
over a circle of large radius centered at $0$. Since the integrand is
small as $|z|\to \infty$ the answer must go to zero as the radius goes to infinity.
But inside the circle there are poles at $z=0$ and $z= \pm k$ for $k$ fr... | 14 | https://mathoverflow.net/users/38624 | 140593 | 76,813 |
https://mathoverflow.net/questions/138160 | 4 | I have the following problem:
I need to evaluate the integral $$\int\_{\cos(\alpha)}^{1} P\_l(t)P\_{l'}(t) dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P\_l$ is the l-th Legendre polynomial.
The thing is that this integral occurs in a double series with truly messy functions $f(l)$ and ... | https://mathoverflow.net/users/nan | How to get an expression for this integral(Numerically/Analytically) | It seems what you want is formula (50) here:
<http://mathworld.wolfram.com/LegendrePolynomial.html>
| 6 | https://mathoverflow.net/users/12120 | 140606 | 76,816 |
https://mathoverflow.net/questions/140610 | 4 | Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu\_n$ such that, for every bounded continuous function $f:X\to\mathbb{R}$ we have
$$\int\_X fd\nu\_n\longrightarrow \int\_X fd\nu$$
for some finite measure $\nu$.
By the Lebesgue decomp... | https://mathoverflow.net/users/17111 | Weak continuity of Lebesgue decomposition | The answer is no. For example, you can construct a sequence of $g\_n\in L^1(\mathbb{R}^n)$ converging to the Dirac $\delta$ measure.
Furthermore, we can also construct a sequence of singular measures converging to an $L^1$ function, e.g. Dirac $\delta$ measures suppored on finitely many points (with suitable weights... | 5 | https://mathoverflow.net/users/22238 | 140613 | 76,818 |
https://mathoverflow.net/questions/140603 | 5 | **Background** Because a bounded distributive lattice can be represented by the clopen sets of a Priestley space, I tried to learn some basics about Priestley spaces. After reading (on Wikipedia)
>
> A Priestley space is an ordered topological space with special properties.
>
>
>
I googled for "ordered topolog... | https://mathoverflow.net/users/20781 | How to define compatible topology for first-order structures? | **Many notions of compatibility between a partially ordered set and a topology on its underlying set are analogous to separation axioms and other well known concepts from general topology.**
One can generalize the separation axioms and other notions such as $T\_{2}$,complete regularity, and zero-dimensionality to axi... | 4 | https://mathoverflow.net/users/22277 | 140625 | 76,820 |
https://mathoverflow.net/questions/140622 | 14 | Assume $G$ is a connected locally compact group and $M$ is a maximal compact subgroup of $G$. Is $M$ connected too?
| https://mathoverflow.net/users/nan | Are maximal compact subgroups of connected groups connected? | **Disclaimer:** Locally compact groups are absolutely not my field of expertise. I hope an expert can check my statements below, and perhaps add some details and references.
The Malcev–Iwasawa theorem implies that any connected, locally compact group $G$ satisfies:
* $G$ has a maximal compact subgroup;
* there exis... | 19 | https://mathoverflow.net/users/21095 | 140638 | 76,824 |
https://mathoverflow.net/questions/140592 | 15 | Let $K$ be a $p$-adic field and $\chi : Gal\_K \rightarrow \mathbb{Q}\_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I\_K)$ is finite (by Sen's theory), and that it is Hodge-Tate of weight $k$ iff $\chi.\chi\_p^{-k}$ is HT of weight $0$.
Is there a similar description for De Rham,... | https://mathoverflow.net/users/39091 | What is the classification of characters in $p$-adic Hodge theory? | The de Rham characters are the same as the Hodge-Tate ones. The semistable ones are the same as the crystalline ones, and in your notation they are the de Rham ones for which $(\chi \cdot \chi\_p^{-k})(I\_K)$ is trivial (and not merely finite). This can be found for example in Fontaine and Mazur's paper.
| 18 | https://mathoverflow.net/users/5743 | 140640 | 76,825 |
https://mathoverflow.net/questions/140643 | 2 | I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$):
$$
\gamma := \|p - p'\| = \sum\_{x\in X}|p(x) - p'(x)|.
$$
Similarly to [this paper](http://www.gatsby.ucl.ac.uk/~gretto... | https://mathoverflow.net/users/11768 | Empirical estimator fot the total variation distance on a finite space | Since your state space is finite, you will have that $\|p\_n-p\|\to 0$ and $\|p\_n'-p'\|\to 0$ at exponential rate of decay of probability (simply from finite alphabet large deviations - for example, use section 2.1 in Dembo-Zeitouni's large deviations book). That is,
$P(\|p\_n-p\|>\delta)\leq n^{|S|} e^{-n I(\delta)}$... | 3 | https://mathoverflow.net/users/35520 | 140649 | 76,828 |
https://mathoverflow.net/questions/140578 | 6 | Let $X$ be a smooth variety over $\mathbb{C}$ and $\mathscr{A}$ a sheaf of twisted differential operators on $X$. The latter comes equipped with a natural filtration and the associated graded algebra $\text{gr } \mathscr{A}$ is identified with $\text{Sym } \mathscr{T}\_X$, the sheaf of functions on the cotangent bundle... | https://mathoverflow.net/users/3544 | Poisson structure on the cotangent bundle | Both of these constructions are étale local on $X$. I'm using étale since you seem to want to work in algebraic geometry; it's probably better to say it's local in the classical/analytic topology.
In either case, I only need to show this for $\mathbb{C}^n$. In that case, it's simply a matter of writing things down; m... | 2 | https://mathoverflow.net/users/66 | 140660 | 76,832 |
https://mathoverflow.net/questions/140654 | 4 | In Birkenhake and Lange's book, they prove a version of the Nakai-Moishezon theorem for complex abelian varieties that says that if $L\_0$ is an ample line bundle on a complex abelian variety $X$ of dimension $g$, then a line bundle $L$ is ample if and only if $(L^\nu\cdot L\_0^{g-\nu})>0$ for $\nu=1,\ldots,g$ (Corolla... | https://mathoverflow.net/users/14143 | Nakai-Moishezon theorem for abelian varieties | On an abelian variety (regardless of the characteristic), an effective divisor with positive self-intersection is ample. To be more precise, it suffices here to recall that on any **simple** abelian variety, all non-zero effective divisors are ample; apply this to the factors in a decomposition (mod isogenies) of an ar... | 11 | https://mathoverflow.net/users/26522 | 140665 | 76,834 |
https://mathoverflow.net/questions/140644 | 6 | The well known result of Erdős, states that
>
> Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$
>
>
>
What I am wondering is
>
> When can we expect equality to hold? I.e for which parameters $(g,k)$ do we have graphs with girth $g$ and chromatic numb... | https://mathoverflow.net/users/1737 | Minimal graphs of prescribed girth and chromatic number | There are no more pairs (g,k). Indeed, one can start with a graph of large chromatic number and large girth. Deleting vertices one at a time, one gets a subgraph with chromatic number exactly k and girth at least g. As long as k>2, adding a disjoint cycle of length g keeps the chromatic number k and the girth will be g... | 10 | https://mathoverflow.net/users/39117 | 140668 | 76,836 |
https://mathoverflow.net/questions/140655 | 9 | what are the examples of elliptic curves defined over $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?
| https://mathoverflow.net/users/30999 | Supersingular elliptic curves over $\mathbb{Q}$ | In fact, this cannot happen: an elliptic curve over $\mathbb{Q}\_p$ is supersingular if and only if its associated mod $p$ Galois representation is irreducible, but if it is irreducible as a representation of $\mathbb{F}\_p[G\_{\mathbb{Q}\_p}]$ then it is certainly irreducible as a representation of $\mathbb{F}\_p[G\_{... | 22 | https://mathoverflow.net/users/2481 | 140669 | 76,837 |
https://mathoverflow.net/questions/140659 | 4 | Let $\pi, \pi'$ be a unitary, irreducible, supercuspidal representations of $GL\_2(F)$. Does an equality of roots numbers $\epsilon(\pi, \psi, s) = \epsilon(\pi', \psi, s)$ for all $s \in \mathbb{C}$ imply an isomorphism $\pi \cong \pi'$?
| https://mathoverflow.net/users/10400 | Are supercuspidal reps of GL(2) uniquely determined by the rootnumber | No -- there are far too many supercuspidals $\pi$ for it to be possible to distinguish them by a single $\varepsilon$-factor -- but this is true if you also consider root numbers of twists; this is called the "local converse theorem" and is in the Jacquet-Langlands book (SLN 114). See also [Andrew Snowden's thesis](htt... | 8 | https://mathoverflow.net/users/2481 | 140671 | 76,838 |
https://mathoverflow.net/questions/140674 | 5 | Let $G$ be a connected Lie Group and $K<G$ a maximal compact subgroup.
Denote by $\Omega^q(G/K)^G$ the $G$-invariant real-valued $q$-forms on the manifold $G/K$, i.e. those forms $\omega$ s.t. $g^\*\omega=\omega$, where $g$ denotes the left translation mapping $hK$ to $(gh)K$.
Evaluation at the identity $eK$ yield... | https://mathoverflow.net/users/39270 | G-invariant differential forms on homogeneous space of Lie Groups | It should be $\operatorname{Hom}(\Lambda^q(\mathfrak{g}/\mathfrak{k}),\mathbb{R})^K$, invariant under $K$. It isn't just the $\mathbb{R}$-linear stuff. Imagine $G$ is the rotation group of the sphere, $K$ the subgroup fixing the north pole. Then to be invariant under $G$, you need to invariant under all transformations... | 4 | https://mathoverflow.net/users/13268 | 140680 | 76,841 |
https://mathoverflow.net/questions/140647 | 3 | The ihara zeta function of a graph $X$ is defined as
$$\zeta\_X(u)=\prod\_{ [C] }(1-u^{v(C)})$$
where the product is over the primes of the graph( A.Terras Zeta functions of graphs a stroll through the Garden)
The question is : Can we take two graphs with the same ihara but different multiset of lengths of primes?
... | https://mathoverflow.net/users/14726 | Primes and ihara zeta function on graphs | No, this can't happen. You would have
$$
1=\prod\_{n}(1-u^n)^{k\_n},
$$
where $k\_n\in\mathbb Z$ is the difference of the number of primes in the first graph of length $n$ minus the number in the second graph.
Applying the logarithm you get
$
0=\sum\_{m}c\_m u^m,
$
where
$$
c\_m=-\sum\_{d|m}dk\_d/m.
$$
Which implies ... | 3 | https://mathoverflow.net/users/nan | 140682 | 76,842 |
https://mathoverflow.net/questions/140681 | 5 | The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?
| https://mathoverflow.net/users/37661 | Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization? | Indeed, the quantization of angular momentum and of spin is an example of the integrality condition in the [geometric quantiation](http://ncatlab.org/nlab/show/geometric+quantization) of the 2-sphere, regaded as a symplectic phase space with its canonical volume form taken as the symplectic form. This is spelled out in... | 5 | https://mathoverflow.net/users/381 | 140685 | 76,843 |
https://mathoverflow.net/questions/140628 | 35 | I have been trying to write up some notes on completion of ordered fields, ideally in the general case (i.e., not just completing $\mathbb{Q}$ to get $\mathbb{R}$ but considering the completion via Cauchy sequences of any ordered field). I have found the technical details of this to be surprisingly thorny, especially c... | https://mathoverflow.net/users/1149 | On the universal property of the completion of an ordered field | The statement can be corrected by adding one word:
>
> Theorem 8.7.1: Let $K$ be an ordered field. Then there is a complete ordered field $\tilde{K}$ and a dense order-embedding $\lambda:K \to \tilde{K}$ such that to each **cofinal** order-embedding $f:K \to L$ into a complete ordered field $L$ there is a unique or... | 13 | https://mathoverflow.net/users/2000 | 140689 | 76,844 |
https://mathoverflow.net/questions/140691 | 6 | Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?
Is it true that $\pi\_1(M)=0$? (*sorry this first bit of the question is silly since any $S^k$ can appear*) Is it true that $M$ has to be homeomorphic to a sphere? If not, ... | https://mathoverflow.net/users/13441 | Fixed component of an $S^1$ action on $S^n$ | The fixed point set need not be simply connected in general. If $M$ is any smooth homology $(n-2)$-sphere that bounds a smooth contractible $(n-1)$-manifold $W$ (such exist in abundance), then $S^1$ acts smoothly on the $(n+1)$-disk $W\times D^2$, by rotations in the $D^2$ factor, with fixed point set $W$ and therefore... | 9 | https://mathoverflow.net/users/1822 | 140692 | 76,845 |
https://mathoverflow.net/questions/140656 | 4 | Let $\mathcal C$ be a category and suppose $\cal B \subseteq C$ is a full subcategory. Let $i \colon \mathcal B \longrightarrow \cal C$ denote the inclusion functor. Suppose that $S \subseteq \operatorname{Mor}\mathcal C$ is a class of morphisms in $\mathcal C$. Then we get a functor
$$\tilde i \colon \mathcal B[(S ... | https://mathoverflow.net/users/38418 | Localisation of inclusion functors | Let $\mathscr{C}$ a category and let $\Sigma \subset \mathscr{C}$ be a wide subcategory (i.e. closed under composition and containing identites).
There exists (generally in a more large sets universe) the category of fractions $P: \mathscr{C} \to \mathscr{C}(\Sigma )$, and $P$ is the identity map on objects. Given a ... | 1 | https://mathoverflow.net/users/6262 | 140698 | 76,846 |
https://mathoverflow.net/questions/139685 | 5 | * I think the statement is that for any dimensional CFT the following is true,
$$\langle T^{\mu}\_\mu \rangle = \sum B\_n I\_n - 2(-1)^{d/2}AE\_d,$$
where $E\_d$ is the `"Euler density" and $I\_n$ are the independent "Weyl invariants of weight $-d$".
(...I am not sure of the definition of the geometric quantities... | https://mathoverflow.net/users/36554 | Proof of the general expression for anomaly in a CFT and its partition function | The answer to your question is contained in the recent book by Spyros Alexakis ``The Decomposition of Global Conformal Invariants'' published last year by Princeton University Press. He answers the Deser-Schwimmer conjecture which (as far as I can understand your question) is the broader context of what you are asking.... | 3 | https://mathoverflow.net/users/17969 | 140701 | 76,848 |
https://mathoverflow.net/questions/140703 | 7 | Is it possible to construct a smooth action of $S^1\times S^1$ on $S^{2n+1}$ ($n\ge 2$) such that no point on $S^{2n+1}$ has an infinite stabilizer?
Note that if such an action exists, it can not be linear.
| https://mathoverflow.net/users/13441 | Pseudofree $T^2$ actions on spheres | There is a bit of a disconnect between the title and the actual question. Usually a semifree action is one in which the only isotropy groups are the trivial group and the whole group. The actions with only finite isotropy groups, in the body of your question, are often called ``pseudofree'' actions. If a torus were to ... | 9 | https://mathoverflow.net/users/1822 | 140707 | 76,850 |
https://mathoverflow.net/questions/140531 | 9 | Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert polynomial $h$ is of finite type (even quasi-projective).
This means, in particular, that the moduli space has only finitely m... | https://mathoverflow.net/users/39201 | There are only finitely many varieties up to deformation | I believe the two things you are relating:
The finiteness of the
* number of deformation types
* number of components of the moduli space
of polarized varieties are essentially equivalent problems.
The way moduli spaces are usually constructed is that first one finds a projective space that contains all the ob... | 7 | https://mathoverflow.net/users/10076 | 140710 | 76,851 |
https://mathoverflow.net/questions/140711 | 1 | Are there some good books on Green function (or approximation Green function) and its application(mainly used in PDEs)? Any reply will be appreciated!
| https://mathoverflow.net/users/35338 | Good books on Green function? | For wave equations, a classic book by Gerard Friedlander is ``The wave equation on
curved spacetime'', which describes the Hadamard parametrix method.
There is another nice recent book by Christian Baer called Linear wave equations
on Lorentzian manifolds.
| 3 | https://mathoverflow.net/users/17969 | 140717 | 76,855 |
https://mathoverflow.net/questions/140713 | 3 | Given $c<\infty$ colors, positive integers $k\_1,\dots,k\_n$ and positive integers $N\_1,\dots,N\_n$. Then there exist positive integers $M\_1,\dots,M\_n$ so that for disjoint finite sets $A\_1,\dots,A\_n$ of cardinalities $|A\_i|=M\_i$, $1\leq i\leq n$, the following statement holds:
assume that each array $(B\_1,\d... | https://mathoverflow.net/users/4312 | Multipartite Ramsey theorem | This is indeed well known, although I can't give a reference where it is stated in this exact form. If I see right, Theorem 49 of Erdos-Rado: A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427--489 (available as <http://www.renyi.hu/~p_erdos/1956-02.pdf>) is at least very close to it.
R. Rado: D... | 2 | https://mathoverflow.net/users/6647 | 140722 | 76,857 |
https://mathoverflow.net/questions/140718 | 2 | We know that a higher-order ode can be converted to dynamical system by replacing each higher-order derivative by a new variable. What about inverse problem? Does a dynamical system convert to a higher-order ode?
| https://mathoverflow.net/users/39246 | Is autonomous dynamical system equivalent to one single higher-order ode? | For example, consider a system such as
$$ \dot{x} = x, \dot{y} = y $$
for which the origin is a proper node. There is no autonomous second-order ODE
$\ddot{z} = f(z, \dot{z})$ that has a proper node: if the linearization at an equilibrium point has a double eigenvalue, it is an improper node. So there
can be no smooth... | 5 | https://mathoverflow.net/users/13650 | 140723 | 76,858 |
https://mathoverflow.net/questions/140637 | 10 | Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the fibers (which are complex curves) are at least $2$. By abuse of notation I call $X$ a Kodaira fibered surface. What is ... | https://mathoverflow.net/users/37808 | Uniformization of Kodaira fibered surfaces | Here are the arguments to exclude polydisk and the ball (there are no other complex 2-dimensional bounded symmetric domains: In fact, one can do without this and argue that any domain other than the ball would have rank $\ge 2$ and, hence, Margulis superrigidity theorem would apply).
1. Kefeng Liu ("Geometric height... | 9 | https://mathoverflow.net/users/21684 | 140725 | 76,859 |
https://mathoverflow.net/questions/140735 | 0 | Are there any algebraic irrational numbers in $\{log\_xy|x,y\in\mathbb{N},x,y\geq2\}$?
| https://mathoverflow.net/users/39304 | Equation in integers of irrational degree | <http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem>,
Suppose your set produces an algebraic irrational number
$ \log\_xy = z,$ then $ x^z = y $ is transcendental by the theorem . But $y$ is a natural number, which are algebraic by nature. Thus we have arrived at a contradiction.
| 3 | https://mathoverflow.net/users/nan | 140737 | 76,860 |
https://mathoverflow.net/questions/139432 | 21 | For the notations I am using, I refer to the Appendix at the end of this post.
Here is what, for the sake of this post, I consider to be Reifegerste's theorem:
**Theorem 1.** Let $n\in\mathbb N$ and $i\in\mathbb N$. Let $\sigma$ and $\tau$ be two permutations in $S\_n$ such that $\sigma$ and $\tau$ differ by a Knut... | https://mathoverflow.net/users/2530 | Has Reifegerste's Theorem on RSK and Knuth relations received a slick proof by now? | First, just for clarity about the question, the [Reifegerste preprint](http://arxiv.org/abs/math/0309266v1) dates from September 2003, her [paper](http://www.ams.org/mathscinet-getitem?mr=2061380) was published in 2004, and Jacob Post's thesis is from 2009.
But the theorem is easy to show from things known well befor... | 16 | https://mathoverflow.net/users/19077 | 140739 | 76,861 |
https://mathoverflow.net/questions/140639 | 7 | Let $C$ be an $(\infty,1)$-topos. The $(\infty,1)$-category of group objects in $C$ is a full sub-$(\infty,1)$-category of groupoid objects in $C$:
$${\mathsf{Grp}}(C) \hookrightarrow {\mathsf{Grpd}}(C)$$
Is this full subcategory reflective? Here is one way to go about constructing a $(\infty,1)$-functor in the other... | https://mathoverflow.net/users/nan | Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects? | If $\mathcal{C}$ is an $\infty$-topos, the $\infty$-category of groupoid objects of $\mathcal{C}$ is equivalent to the full subcategory of $Fun( \Delta^1, \mathcal{C})$ spanned by the effective epimorphisms $X \rightarrow Y$. Under this equivalence, the group objects correspond to the full subcategory where $X$ is a fi... | 11 | https://mathoverflow.net/users/7721 | 140742 | 76,864 |
https://mathoverflow.net/questions/140730 | 0 | Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth and bounded function which tends to 0 at infinity. Define, for $t>0$, the distribution
$$
\nu (t) = \int \limits \_{f(x)\ge t} dx,
$$
and (in the distributional sense) the positive measure
$$
\mu = -\frac{d\nu }{dt}.
$$
By a change of variables we see that
$$
\langle \mu... | https://mathoverflow.net/users/19433 | Why equality of singular supports? | $\nu$ is a distribution on the real line and the operator $P=d/dt$ is elliptic with constant coefficients. In that case we have
$$
\text{singsupp $\nu$}=\text{singsupp $P\nu$}
$$
for the $C^\infty$ singular support as well as for the analytic singular support. The same equality holds for wave-front-sets (smooth and ana... | 3 | https://mathoverflow.net/users/21907 | 140743 | 76,865 |
https://mathoverflow.net/questions/140724 | 2 | A $2$-ary predicate $R$ in [Grzegroczyk-hierarchy](http://en.wikipedia.org/wiki/Grzegorczyk_hierarchy) is binary-valued function $R\colon\mathbb{N}^2\to\{0,1\}$. We say $R$ encodes function $f:\mathbb{N}\to\mathbb{N}$ if $R(x,y)$ is true if and only if $f(x)=y$.
Let us assume that $i>2$ and $j>i$. Is there a functio... | https://mathoverflow.net/users/nan | Predicates encoding functions in Grzegorczyk-hierarchy | The general principle is that the *faster* a function grows, the *easier* it is to compute its graph, because the extremely long output of the function is given to us as an input, which provides immense computational power.
For the specific question, it is well known that e.g. graphs of all kinds of variants of the A... | 5 | https://mathoverflow.net/users/12705 | 140744 | 76,866 |
https://mathoverflow.net/questions/140754 | 14 | Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection
with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?
If the answer is *yes*, can $P$ be chosen in such way that there is a constant $C\_P$ with
the property that for every convex subset $S \subset \ma... | https://mathoverflow.net/users/28104 | Sets of evenly distributed points in the Euclidean plane | There is a set $P$. For the construction of this set first take the squares of area $1$ whose edges are integers and numerate them. For each square, say $S\_n$, you can take a square lattice in it such that any convex inside the square that do not intersect the lattice has area less than $a\_n$ for any $a\_n>0$, just t... | 16 | https://mathoverflow.net/users/34575 | 140757 | 76,869 |
https://mathoverflow.net/questions/140753 | 0 | Suppose you have 4 matrices with singular value decompositions
$A = U\_1 \Sigma\_A V\_1^{\dagger}$, $B = U\_2 \Sigma\_B V\_2^{\dagger}$, $C = U\_1 \Sigma\_C V\_1^{\dagger}$ and $D = U\_2 \Sigma\_D V\_2^{\dagger}$ such that $\Sigma\_A \Sigma\_C$ and $\Sigma\_B \Sigma\_D$ are both nonzero.
Are the singular value decomp... | https://mathoverflow.net/users/39313 | does the basis in the singular value decomposition of a sum depend on the singular values of the summands | no, just take as a counterexample: $V\_1=U\_1$, $V\_2=U\_2$, $\Sigma\_A=\mathbb{1}$, $\Sigma\_D=\mathbb{1}$, so $A+B=U\_2(\Sigma\_B+\mathbb{1})U\_2^{\dagger}$, $C+D=U\_1(\Sigma\_C+\mathbb{1})U\_1^{\dagger}$, so your assumption fails unless $U\_1=U\_2$.
| 1 | https://mathoverflow.net/users/11260 | 140759 | 76,871 |
https://mathoverflow.net/questions/140763 | 6 | I have been reading a bit about Zhang's proof and the associated Polymath8 project.
Though Tao's high level summary
<http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/>
is interesting it still is rather technical. So without any knowledge of the techniques I am unable to g... | https://mathoverflow.net/users/3757 | Other implications of Zhang's method | Zhang's strategy shows that any admissible tuple of size h contains at least 2 primes infinity often, as long as h is larger than some threshold (the primary theoretical thrust of the polymath project has been to reduce the required value of h). This approach seems unable to give (1), but immediately gives (2).
| 9 | https://mathoverflow.net/users/630 | 140764 | 76,872 |
https://mathoverflow.net/questions/140728 | 1 | I have a little problem to fully understand the next thing:
>
> Let $F:\mathbb{P}^2 \dashrightarrow \mathbb{P}$ a rational map. Let
> $\sigma:\bar{\mathbb{P}}^2 \rightarrow \mathbb{P}^2$ be a composition
> of $\sigma$-processes resolving the indeterminacy points of the
> rational map $F$, so that $\bar{F} = F\ci... | https://mathoverflow.net/users/32645 | Fibers of the resolution of indeterminacy points of a rational map | It seems to me that the only way this can happen is if $L\_\infty\subset \mathbb P^2$ maps to $\infty\in\mathbb P^1$.
So, $\overline Y\_\infty$ consists of $\sigma^{-1}\_\*L\_\infty$ (a.k.a., the strict transform of $L\_\infty$) and possibly a few $\sigma$-exceptional curves that map to $L\_\infty$ via $\sigma$.
Th... | 0 | https://mathoverflow.net/users/10076 | 140769 | 76,876 |
https://mathoverflow.net/questions/140761 | 4 | I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a link of the paper <http://www.stat.yale.edu/~hz68/619/Brown1971.pdf>. One of the main ideas of the paper is to transfer th... | https://mathoverflow.net/users/39323 | diffusions corresponding to estimators | Here is an attempt to answer your questions, admitting that I have not read the paper carefully.
The generator corresponding to a diffusion satisfying the SDE $dX\_t = b(X\_t) dt + \sigma(X\_t) dB\_t$ is the following partial differential operator $A$, acting on a function $f$ as follows
$$
Af(x) = \sum\_i b\_i(x) \,... | 2 | https://mathoverflow.net/users/36687 | 140771 | 76,877 |
https://mathoverflow.net/questions/138560 | 7 | What are current trends/questions in algebraic logic? I mean the research developed by Paul Halmos.
Could anyone give some references for the overview of its history? Any overview of its application to computer science and computability theory is welcome.
| https://mathoverflow.net/users/14024 | What are current trends/questions in algebraic logic? | I ran across this today, and was reminded of this question: I think the article "[Algebraic logic, Where does it stand today?](https://www.jstor.org/stable/3396712)" by T. S. Ahmed addresses your question.
| 7 | https://mathoverflow.net/users/8133 | 140779 | 76,881 |
https://mathoverflow.net/questions/140535 | 2 | A ring $R$ (with 1) has IBN property if free $R$-modules have unique rank (e.g., commutative rings). In the same fashion, lets call $R$ a **good** ring if in every free $R$-module any independent set can be extended to a basis (not every commutative ring has this property). I have two questions:
1. Is the class of *... | https://mathoverflow.net/users/nan | A class of rings related to rings with IBN property | The Rings you are calling good, are called (left or right) Steinitz. They are characterized as: **A ring is (left) right Steinitz if and only if it is (left) right perfect and local**. This means that every (left) right Steinitz ring has IBN property. You may like to see the following
**[1]** Chew, Neggers, On the ex... | 4 | https://mathoverflow.net/users/nan | 140785 | 76,883 |
https://mathoverflow.net/questions/140786 | 5 | The Levy-Solovay theorem says that if $\kappa$ is measurable, then it remains measurable in the extension by a small forcing ($|\mathbb{P}|<\kappa$). Is still true if we replace $|\mathbb{P}|<\kappa$ with "$\mathbb{P}$ has the $\lambda-\textrm{c.c.}$ for some $\lambda<\kappa$"? Or even, can a c.c.c. forcing destroy mea... | https://mathoverflow.net/users/38814 | Generalization of Levy-Solovay theorem to kappa-c.c. forcings | The forcing to add $\kappa$ many Cohen reals is c.c.c and thereby preserves all cardinals but destroys the fact that $\kappa$ is even strongly inaccessible. (See for example Kunen "Set Theory: An Introduction to Independence Proofs" North-Holland, Ch. VII.)
| 10 | https://mathoverflow.net/users/6942 | 140787 | 76,884 |
https://mathoverflow.net/questions/140770 | 7 | There are several ways of producing manifolds,say:
1.orbits space of group action
2.connected sum of manifolds
3.underlying topological space of nonsingular algebraic set
....
here,i am interested in the 3rd one.
A well known theorem due to Nash and Tognoli says that Every compact smooth manifold is diffeom... | https://mathoverflow.net/users/39332 | smooth manifolds as real algebraic set (continued) | It isn't too easy to find good equations for surfaces as algebraic subspaces of $\mathbb{R}^3$, but if you are willing to use $\mathbb{R}^n$ for larger $n$ then the picture is clearer. There are standard ways in algebraic geometry to produce surfaces as complex subvarieties $X\subset\mathbb{C}P^2$. If we let $U$ denote... | 8 | https://mathoverflow.net/users/10366 | 140799 | 76,886 |
https://mathoverflow.net/questions/140829 | 21 | I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on is applications of finite groups in statistics and data analysis. Unfortunately, the only book I know on the subject is P... | https://mathoverflow.net/users/66 | Easier reference for material like Diaconis's "Group representations in probability and statistics" | I have a chapter on this in my book [Representation theory of finite groups](https://doi.org/10.1007/978-1-4614-0776-8). Sorry for the self promotion. It is intended for advanced undergrads. I basically focus on the abelian case, giving the upper bound lemma on convergence rates and the description of the eigenvalues f... | 14 | https://mathoverflow.net/users/15934 | 140830 | 76,898 |
https://mathoverflow.net/questions/140762 | 1 | Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H\_0^1(\Omega)^n$) and suppose $B$ is a Banach space that is continuously and densely embedded in $H$. Let $g\geq 0$ and continuous, is the set $$C(B,g)... | https://mathoverflow.net/users/26827 | Continuous and dense embeddings and the density of sets in Hilbert space | Here is a counterexample. Take some strictly positive function $w\colon [0,1] \to R$ which is in $L^2$ and has a dense set of singularities, for example
$$
w(x) = \sum\_{q \ge 1}\sum\_{p=1}^q {1\over q^4 |x-p/q|^{1/4}}
$$
Then I choose $B$ to be the space of all functions of the form $f = Fw$ with $F$ continuous and $... | 4 | https://mathoverflow.net/users/38566 | 140831 | 76,899 |
https://mathoverflow.net/questions/140837 | 0 | There are at least two ways people look at statistical data:
A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, variance, and other central moments, maybe characteristic functions.
B. Everybody else would consider median and other p... | https://mathoverflow.net/users/38448 | Relating percentiles to moments | Question 1. It is impossible to give any estimate without extra assumptions.
Take a random variable which takes only two values: $a>0$ and $b<0$, both with probability
$1/2$. The median is zero (see the remark below).
All moments can be arbitrarily large or arbitrarily small: to make them large choose
$a$ very large, ... | 1 | https://mathoverflow.net/users/25510 | 140845 | 76,907 |
https://mathoverflow.net/questions/140849 | 10 | Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$.
It seems that the solution of linear ODE
$$\dot x= Ax$$
can not be written in a closed form using $\int$, $A$, $x(0)$ and the standard functions.
>
> **Question 1.** Is it a theorem?
>
>
>
If "no".
>
> **Question 2.** Any idea... | https://mathoverflow.net/users/10330 | Solution of linear ODE | This is known as Differential Galois Theory, first developed by Picard and Vessiot. In your case you should look for authors such as Kolchin or Singer and Van Der Put. Some systems definitely admit solutions in "closed form" (you can build them!), but most won't.
The ingredient is the "monodromy group", measuring the ... | 19 | https://mathoverflow.net/users/24309 | 140851 | 76,910 |
https://mathoverflow.net/questions/140850 | 2 | Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?
| https://mathoverflow.net/users/27398 | elementary question on a completion of a ring | If the characteristic of $k$ is not $2$, $(t^2+\epsilon)=(t+\epsilon/2)^2$. So write $t'=t+\epsilon/2$. Them we are looking at the $t'^2$-adic completion of $k[\epsilon][[t']]$.
If the characteristic is $2$, then $(t^2+\epsilon)^2=t^4$. So this is just the $t$-adic completion.
Alternately, you can observe that a se... | 7 | https://mathoverflow.net/users/18060 | 140852 | 76,911 |
https://mathoverflow.net/questions/140731 | 5 | While it is true that $\mathcal P(\kappa)$ is a complete Boolean algebra, it is not necessarily true that $\mathcal P(\kappa)/I$ is complete for an ideal $I$. In particular if we consider $I=J\_{bd}$ the ideal of bounded subsets of $\kappa$.
For example, Hausdorff showed that there is an $(\omega\_1,\omega\_1)$ gap i... | https://mathoverflow.net/users/7206 | How complete is $\mathcal P(\kappa)/J_{bd}$? | If $\kappa$ is regular, then $\mathcal{P}(\kappa)/J\_{bd}$ is a $\kappa$-complete boolean algebra. If $\langle A\_\alpha : \alpha < \delta < \kappa \rangle$ is a sequence of subsets of $\kappa$, then the union of these is a least upper bound. This uses the $\kappa$-completeness of $J\_{bd}$. It is not $\kappa^+$-comple... | 6 | https://mathoverflow.net/users/11145 | 140853 | 76,912 |
https://mathoverflow.net/questions/140856 | 5 | As is common terminology in graph reconstruction, given a graph $G$, we call a vertex deleted subgraph of $G$, a *card*, and call the multiset of all cards, the *deck* of $G$. The graph reconstruction problem is to determine the isomorphism type of $G$ by only looking at its deck.
**Question:** Is there a slick argum... | https://mathoverflow.net/users/2384 | Reconstructing the number of Hamiltonian cycles | Bill Kocay found a more direct combinatorial method to reconstruct the number of hamiltonian cycles and some other spanning subgraphs. It is in his paper "Some new methods in reconstruction theory", Lecture Notes in Mathematics Volume 952, 1982, pp 89-114. You can get it [here](http://link.springer.com/content/pdf/10.1... | 7 | https://mathoverflow.net/users/9025 | 140860 | 76,914 |
https://mathoverflow.net/questions/140859 | 16 | [Kolmogorov superposition theorem](https://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&sqi=2&ved=0CDQQFjAB&url=http://wissrech.ins.uni-bonn.de/research/pub/braun/remonkoe.pdf&ei=tzAhUvu_A-fCsAS78ID4BA&usg=AFQjCNF1XfRRDuAqJ6_7C60ttGO81rmoQw&sig2=SOHrvWHqgaCrAKN1WO3HGg&bvm=bv.51495398,d.cWc) states tha... | https://mathoverflow.net/users/5506 | Kolmogorov superposition for smooth functions | The answer is no. There exist analytic functions $f$ of 3 variables such that they cannot be represented as a composition of continuously differentiable functions of two variables. This is old result of Vitushkin. You can find nice story of Hilbert's thirteen problem in Vitushkin, A. G. Hilbert's thirteenth problem and... | 21 | https://mathoverflow.net/users/1811 | 140862 | 76,916 |
https://mathoverflow.net/questions/140863 | 1 | Let $\left\{\mathbf{p}\_1,\dots, \mathbf{p}\_k\right\}$ be a set of points in $n$-dimensional Euclidean space, and let the second moment of these points be defined as:
$
U=\sum \limits\_{i=1}^{k} ||\mathbf{p}\_i -\bar{\mathbf{p}}||^2,
$
where $\bar{\mathbf{p}}$ is their centroid.
Let two points $i$ and $j$ be co... | https://mathoverflow.net/users/39213 | Maximum Dispersion of a Connected Geometric Graph | Notice that this is the same as maximizing $\sum\_{i<j} |p\_i-p\_j|^2$. Also by rescaling assume that $\lambda =1$. Since your graph is connected there is a spanning tree whose vertices are the $p\_i$'s. Let $\text{d}(i,j)$ denote the graph theoretic distance between $p\_i$ and $p\_j$ along this tree. From the triangle... | 2 | https://mathoverflow.net/users/2384 | 140864 | 76,917 |
https://mathoverflow.net/questions/140867 | 5 | I'm looking for a particular description of the Hopf algebra structure on the ring of quasisymmetric functions.
Let me illustrate by giving this kind of description for the Hopf algebra of symmetric functions.
Fix a ground field $k$.
**Edit**: I'm happy to assume $k=\mathbb{Q}$. As darijgrinberg pointed out in th... | https://mathoverflow.net/users/5263 | Hopf algebra structure on the ring of quasisymmetric functions | Not sure that it helps but if I'm not mistaken, the $k$-Hopf algebra $Qsym$ you are looking for is the topological dual completed Hopf algebra $k \langle\!\langle y\_i , i \geq 1 \rangle\! \rangle$ of power series in an infinity of non commutative variables with coproduct $\Delta\_\star (y\_n) = \sum\_{p+q = n} y\_p \o... | 3 | https://mathoverflow.net/users/1985 | 140869 | 76,919 |
https://mathoverflow.net/questions/140872 | 10 | I have just started to read about operads, so this question might be silly.
So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain operad. For example, associative algebras are algebras over the associative operad $\mathcal{As}$, commutative algebra... | https://mathoverflow.net/users/32741 | How universal is operadic approach to studying algebras? | One example is a bialgebra. An operad only has operations taking several inputs and one output, so there is no room for comultiplication. On the other hand this is not a serious obstacle. A small modification of the defining axioms will accommodate also this example: bialgebras are algebras over a properad, rather than... | 15 | https://mathoverflow.net/users/1310 | 140876 | 76,920 |
https://mathoverflow.net/questions/140855 | 4 | $A\_\infty$ operad can be described both in terms of Stasheff polytopes and configuration spaces.$A\_n$ operad can be described as subspace of Stasheff operad described using Stasheff polytope. Is there a model for $A\_n$-operad as a configuration spaces?
| https://mathoverflow.net/users/19186 | A_n operad as configuration spaces | At the risk of saying something stupid I'm promoting my comment to an answer. In terms of the Stasheff polytopes the $A\_n$ operad sits inside the $A\_\infty$ operad as the union of all faces of dimension $\leq n-2$. Another way of saying this is that the Stasheff polytopes are stratified spaces with strata indexed by ... | 2 | https://mathoverflow.net/users/1310 | 140881 | 76,921 |
https://mathoverflow.net/questions/140882 | 0 | Is there any way to compute the number of minimal left ideals of $M\_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?
| https://mathoverflow.net/users/nan | Number of minimal left ideals | The lattice of left ideals of $M\_n(K)$ is in bijection with the subspaces of $K^n$, by sending a left ideal $I$ to the subspace of $K^n$ consisting of all rows of elements of $I$ (observe that, from the left, we can do arbitrary row operations). This bijection is order-preserving, and hence the left ideals of $M\_n(K)... | 6 | https://mathoverflow.net/users/nan | 140887 | 76,924 |
https://mathoverflow.net/questions/140883 | 0 | Let $p\in (0,1)$ be fixed and let $X$ be a binomial random variable with parameters n and p. Consider a related normal random variable $N$ with mean $np$ and variance $np(1-p)$. Is it true that for some $x=x(n)$ we have $P(X>np+x)=(1+o(1))P(N>np+x)$? That is, if true, I would like to know how large $x$ can be. Is it tr... | https://mathoverflow.net/users/24494 | Large deviations for bernoulli sums | There are quite a lot of approximations more accurate than the pure normal distribution, with a large literature difficult to sort out. You can see several in [this paper](http://cs.anu.edu.au/~bdm/papers/littlewood2.pdf) (Adv. Appl. Prob., 21 (1989) 475-478). If I am expanding Theorem 1 correctly, for fixed $p$ that i... | 2 | https://mathoverflow.net/users/9025 | 140888 | 76,925 |
https://mathoverflow.net/questions/140878 | 4 | Can two unitary similar real matrices be orthogonal similar?
suppose $A=U^tBU$ where $U$ is unitary, does there always exists a real orthogonal matrix $O$, such that $A=O^tBO$ ?
| https://mathoverflow.net/users/nan | Can two unitary similar real matrix be orthogonal similar | Unitarily similar real matrices are always orthogonally similar.
The proof can be found in Rached Mneimné and Frédéric Testard's book "Introduction à la théorie des groupes de Lie classiques".
First of all, one sees that two complex square matrices $A$ and $B$ are unitarily similar
if and only if the pairs $(A,A^\s... | 9 | https://mathoverflow.net/users/34951 | 140890 | 76,927 |
https://mathoverflow.net/questions/140807 | 12 | The great dodecahedron is a non-convex regular polyhedron bounded by 12 pentagonal faces, crossing each other, arranged in a star-shaped manner around each of its 12 vertices (see the [Wikipedia page](http://en.wikipedia.org/wiki/Great_dodecahedron) for a picture.) From a topological point of view, the abstract surface... | https://mathoverflow.net/users/39348 | Is there a nice way to "unravel" a great dodecahedron? | No, it is not possible to embed the genus 4 surface in $\mathbb R^3$ with 5-fold axial symmetry. The $2\pi/5$ rotation of the great dodecahedron has four fixed points. Two of these fixed points are at the center of a pentagonal face, and the other two are at a vertex where five pentagons meet. The combinatorial rotatio... | 9 | https://mathoverflow.net/users/284 | 140895 | 76,929 |
https://mathoverflow.net/questions/140480 | 9 | Suppose that $\mathcal{U},\mathcal{V}$ are ultrafilters on sets. Recall that $\mathcal{U}\leq\_{RK}\mathcal{V}$ (here we say $\mathcal{U}$ is Rudin-Keisler less than or equal to $\mathcal{V}$) iff for each first order structure $\mathcal{A}$, the ultrapower $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\ma... | https://mathoverflow.net/users/22277 | Is the product of ultrafilters cancellative? | Convention: I'll use my favorite notation $\mathcal U\otimes\mathcal W$ for what is called $\mathcal U\cdot\mathcal W$ in Jonathan Verner's answer and $\mathcal W\cdot\mathcal U$ in the question. It follows from a theorem of Mary Ellen Rudin (in the 1960's if I remember correctly) that any RK-equivalence between two su... | 8 | https://mathoverflow.net/users/6794 | 140897 | 76,931 |
https://mathoverflow.net/questions/140816 | 3 | It's a bit like asking who [invented the wheel](http://en.wikipedia.org/wiki/Ljubljana_Marshes_Wooden_Wheel), but perhaps there's something out there. This seems beyond the obvious crew: Euclid, Pythagoras, etc. Is there any evidence when this became common knowledge.
Subquestion: Same thing but for geodesics.
| https://mathoverflow.net/users/23064 | Who first realized that the shortest distance between two points is a straight line? | In order to be a question about mathematics this would have to ask not when the fact became common knowledge -- since it is already known to bees and dogs as mentioned -- but when it got expressed as a theorem. In fact there was a traditional objection to the whole idea of proof in geometry by people who said common se... | 6 | https://mathoverflow.net/users/38783 | 140901 | 76,933 |
https://mathoverflow.net/questions/140122 | 3 | A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu\_A$ on $G$ with $\mu\_A(A)=1$. I'm interested in a seemingly stronger condition that naturally connects up $\mu\_A$ for different $A$s.
The condition I want is that one can additionally c... | https://mathoverflow.net/users/26809 | A stronger version of supramenability? | It looks like neat supramenability is equivalent to supramenability, at least given AC. This follows from Proposition 1.7 in the 1989 paper by Armstrong in [this volume](http://books.google.com/books?id=TXhb5wtS8D0C) (page 7). The proof uses the existence of a maximal set $M$ of non-trivial Renyi-ordered finitely addit... | 3 | https://mathoverflow.net/users/26809 | 140909 | 76,938 |
https://mathoverflow.net/questions/140797 | 6 | I have a field $K$ of transcendence degree two over $\mathbb{R}$, and elements $a\_1,a\_2,a\_3\in K$. I would like to understand the set
$$ Q = \{ u\in K^3 : \sum\_i a\_iu\_i^2 = 1\} $$
In particular, I would like to know whether it is nonempty, and if so, I would like to find some examples of elements, and ideally som... | https://mathoverflow.net/users/10366 | Does this quadratic form over a large field represent 1? | Starting from Jason's observations, an obvious obstruction for the existence of a solution would be the existence of a point $M=(y\_1,y\_2)$ in $\mathbb{R}^2$ such that $b\_1(M)>0$, $b\_2(M)>0$, and $b\_4(M)<0$. However, we have
$$16 b\_2=(4y\_1^2)\,b\_4-(5y\_2-2)^2\,b\_1$$
so whenever $b\_1>0$ and $b\_4<0$ we must hav... | 3 | https://mathoverflow.net/users/7666 | 140917 | 76,939 |
https://mathoverflow.net/questions/140912 | 6 | Let $U\_1,U\_2,\ldots$ be iid random variables distributed uniformly on $[0,1]$. I am interested in the random walk $X\_i = \sum\_{j \leq i} U\_j$. In particular,
>
> What is the expected number of points appearing in an interval $[x,x+1]$?
>
>
>
Experimentally, this seems to converge to $2$.
Here is an intu... | https://mathoverflow.net/users/7732 | Random walk with positive uniformly distributed steps | Let $X\_t$ be the number of points in $[0,t]$. Then, $X\_t$ is a renewal process. Let $m(t) = E[X\_t]$. Then [renewal theorem](http://www.randomservices.org/random/renewal/LimitTheorems.html) says
$$
m(t+h) -m(t) \stackrel{t \to \infty}{\longrightarrow} \frac{h}{\mu}
$$
where $\mu = E[U\_1]$ is the expected increment.... | 9 | https://mathoverflow.net/users/36687 | 140918 | 76,940 |
https://mathoverflow.net/questions/140905 | 9 | Let $X$ be a Cohen-Macaulay scheme (let's say of finite type over a field).
Let $X\_{red}$ be the corresponding reduced scheme. Is it true that $X\_{red}$ is also
Cohen-Macaulay?
| https://mathoverflow.net/users/3891 | cohen-macaulayness of reduced and non-reduced schemes | The simplest counter-example I know is the following: [Hartshorne](http://www.jstor.org/stable/2373984) showed that if $k$ has positive characteristic, $k[s^4, s^3t, st^3,t^4]$ (which will be $X\_{red}$) is a set-theoretic complete intersection (said complete intersection will be $X$). The former is well-known to be no... | 8 | https://mathoverflow.net/users/2083 | 140929 | 76,944 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.