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/181575 | 13 | I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles...
>
> First some definitions:
>
>
>
• Let $G$ be compact, simple, and simply connected Lie group, with Dynkin diagram $\Gamma$.
• Let $\{\alpha\_i\}\_{i=1...n}$ be the simple roots ... | https://mathoverflow.net/users/5690 | on the center of a Lie group | I have found a reference that contains the exact statement which I wanted:
It is Theorem 3, on page 14 of [this paper](http://www.sciencedirect.com/science/article/pii/S0001870899919102) of Stephen Sawin.
The vertices of the Weyl alcove which are in bijection with $Z(G)$ are called *sharp corners* (for self-explanato... | 4 | https://mathoverflow.net/users/5690 | 228255 | 106,440 |
https://mathoverflow.net/questions/228159 | 8 | This is question [Selection problem in a collection of non-empty sets](https://mathoverflow.net/questions/228126/selection-problem-in-a-collection-of-non-empty-sets) with a simplification in criterion 3.
Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empt... | https://mathoverflow.net/users/8628 | Edge chromatic number of hypergraphs | This is equivalent reformulation of Erdös-Faber-Lovász conjecture, see Wikipedia page about it.
<https://en.m.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture>
| 5 | https://mathoverflow.net/users/4312 | 228257 | 106,442 |
https://mathoverflow.net/questions/228168 | 9 | [Here](https://mathoverflow.net/questions/219590/why-should-we-believe-in-the-axiom-of-regularity) Professor Blass describes the following cumulative hierarchy of sets:
>
> Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets), then form all sets o... | https://mathoverflow.net/users/85140 | What does the axiom of replacement mean and why should I believe it? | For an argument that the iterative conception implies something weaker than unrestricted Separation (implied by unrestricted Replacement), i.e. $\Sigma\_2$ Replacement, see Randall Holmes 2001 <http://math.boisestate.edu/~holmes/holmes/sigma1slides.ps>. (According to Professor Holmes, “this contain[s] an error, which K... | 4 | https://mathoverflow.net/users/13530 | 228267 | 106,444 |
https://mathoverflow.net/questions/228140 | 30 | When you try to understand the fuss behind the new good categories of spectra that arose on the 90's, you read things such as the following paragraph written by Peter May (from "The Hare and the Tortoise"):
>
> All consumers are now in agreement: Mike's stable homotopy category is
> definitively the right one, up... | https://mathoverflow.net/users/6249 | What is, really, the stable homotopy category? |
>
> I am mildly aware of the fact that there is a stable ∞-categorical universal property. That is certainly interesting, but I would be interested to see how it could be formulated in older language (model categories?)
>
>
>
Symmetric spectra have a model categorical universal property: they form the initial st... | 19 | https://mathoverflow.net/users/85136 | 228270 | 106,445 |
https://mathoverflow.net/questions/228013 | 4 | Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation.
Consider a measurable set $A\subset X$ with $0<\mu(A)<1$
For each $N$ define the sets
$$A\_N=\{x\in X: T^n(x)\in A \forall |n|<N\}$$
By the Birkhoff ergodic theorem $\mu(A\_N)\to 0$ as $N\to \infty$.... | https://mathoverflow.net/users/12395 | Is there a mixing condition to get the decay property I want? | The desired result is false for all mixing systems other than a point:
**Proposition:** let $T$ be an invertible totally ergodic transformation of a standard probability space $(X,\mathcal{F},\mu)$. Then there exists a measurable set $A\subset X$ such that $0<\mu(A)<1$ and
$$\mu\left(\left\{x \in X \colon T^nx \in ... | 7 | https://mathoverflow.net/users/1840 | 228278 | 106,449 |
https://mathoverflow.net/questions/228287 | 8 | Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map
$$
K(\pi,1)\longrightarrow K(\pi,1)/G.
$$
**Question.** Is $K(\pi,1)/G$ an Eilenberg-MacLane space? Does
$$
K(\pi,1)/G=K(\pi \times G,1)?
$$
| https://mathoverflow.net/users/41075 | quotient space of Eilenberg-MacLane space | Suppose a group $H$, not necessarily finite, acts on an Eilenberg-MacLane space $BN$. The homotopy quotient $BN/H$ (which agrees with the ordinary quotient if the action of $H$ is free) fits into a fiber sequence
$$BN \to BN/H \to BH$$
and the long exact sequence in homotopy shows that $BN/H$ has vanishing higher h... | 16 | https://mathoverflow.net/users/290 | 228292 | 106,456 |
https://mathoverflow.net/questions/228285 | 1 | Can someone point me in the direction of a good exposition of the Rees Construction in Hodge Theory?
Thanks!
| https://mathoverflow.net/users/57025 | Looking for a good exposition - Rees Construction | There are many excellent references. It really depends on what you are after. One of the most comprehensive references is the thesis of Olivier Penacchio: <http://arxiv.org/abs/math/0307156>.
| 2 | https://mathoverflow.net/users/439 | 228294 | 106,457 |
https://mathoverflow.net/questions/228293 | 1 | I am maximizing a function $f(x,z)$ on $x$ ($z$ is treated a parameter in the maximization). The function $f$ is strictly concave on both variables.
I know how to use the envelope theorem for the first derivative. But I am interested in knowing if the function $f$ evaluated at the maximum, i.e., $f(x^\*(z),z)$, where... | https://mathoverflow.net/users/85202 | Envelope theorem for second derivative | Yes, the resulting function is concave in $z$.
This operation is sometimes called *partial maximization* and is known to preserve concavity (strict concavity is not required).
More precisely, if $f(x,z)$ is a concave function, then the partially maximized function $g(z)$
$g(z) = \sup\_{x} f(x,z)$
is also concave... | 1 | https://mathoverflow.net/users/1184 | 228313 | 106,465 |
https://mathoverflow.net/questions/228300 | 1 | For every classical r-matrix $r$, there is a Poisson bracket called Sklyanin bracket associated to $r$. It is defined in (3.3) of page 5 in (<http://arxiv.org/pdf/1101.0015v2.pdf>) as follows.
\begin{align}
\{f, g\} = \sum\_{ij} r\_{ij}( \partial\_i^L f \partial\_j^L g - \partial\_i^R f \partial\_j^R g ).
\end{align}
M... | https://mathoverflow.net/users/11877 | Questions about Sklyanin bracket | Perhaps you can use the method of section 4.3 (The second Russian formula (quadratic brackets)) in [Kosmann-Schwarzbach, Lie Bialgebras, Poisson Lie Groups, and Dressing Transformations](http://www.math.polytechnique.fr/cmat/kosmann/lnp2.pdf)? They calculate the bracket for SL(2,R) and SU(2) there.
See also example 4... | 3 | https://mathoverflow.net/users/83246 | 228316 | 106,466 |
https://mathoverflow.net/questions/228039 | 16 | Apologies in advance if this question is inappropriate for MO.
I'm trying to read here and there about $\mathbb A^1$-homotopy theory in algebraic geometry. I understand some abstract machinery is needed, in particular sheaves over sites, some simplicial stuff, and some model categories. The [wiki article](https://en.... | https://mathoverflow.net/users/69037 | (really) basic intuition for $\mathbb A^1$-homotopy theory | Let me try to answer the specific subquestion "What are (intuitively) maps between schemes in the $\mathbb{A}^1$-homotopy category and what does it mean for maps to be homotopic?"
**The general-machinery answer:** computation of maps $f:X\to Y$ in the homotopy category requires a fibrant replacement $\tilde{Y}$ of $Y... | 18 | https://mathoverflow.net/users/50846 | 228317 | 106,467 |
https://mathoverflow.net/questions/227616 | 11 | I am interested in the following simple looking problem on which I am stuck. Let $M$ be a fixed $m$ by $n$ matrix with $\pm1$ elements. Let $x$ and $y$ be two independently sampled random $n$-dimensional vectors whose elements are chosen i.u.d. from $\{-1,1\}$.
>
> Assuming that $m<n$ and both $m$ and $n$ are large... | https://mathoverflow.net/users/45564 | Probability two products are equal | I came across this problem a few days ago, and must say it was a lot of fun to think about. Thank you!
The following is, I believe, a quite complete solution of the problem for a "typical", general $M$, making no assumptions about the relative size of $n$ and $m$ (even $m>n$ is allowed), or about the rank of $M$.
... | 2 | https://mathoverflow.net/users/84637 | 228337 | 106,471 |
https://mathoverflow.net/questions/228259 | 4 | Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is
$H(p,q):=\int\_{\mathbb R}\sqrt{pq}\,dx$.
Let now $p$ be the density of Student's distribution with $d$ degrees of freedom, so that
$$p(x)=C\_d\,(1+x^2/d)^{-(d+1)/2}$$
for rea... | https://mathoverflow.net/users/36721 | Hellinger integral for the Student/Cauchy family | Denote $t=2\tau$. $$
\int\_{\mathbb R}\frac{dx}{(1+x^2)^a\,(1+(x-2\tau)^2)^a}=\int\_{\mathbb R}\frac{dx}{(1+(x+\tau)^2)^a\,(1+(x-\tau)^2)^a}.
$$
We have
$$
\left(1+(x+\tau)^2\right)\,\left(1+(x-\tau)^2\right)=\left(x^2+\tau^2+1\right)^2-4\tau^2x^2=x^4+2x^2(1-\tau^2)+(\tau^2+1)^2=:x^4+Ax^2+B,
$$
so our problem for redu... | 5 | https://mathoverflow.net/users/4312 | 228344 | 106,474 |
https://mathoverflow.net/questions/228335 | 4 | If $X\_i$ is a sequence of $d$ dimensional i.i.d. integer valued random vectors with covariance matrix $\Sigma$ and $\mathbb{E}(X\_i) = \mu$. Let each element of $X\_i$ be chosen i.u.d. from $\{-1,1\}$. We know from the multidimensional central limit theorem that
$$ \frac{1}{\sqrt{n}}\sum\_{i=1}^n (X\_i - \mu)\ \stac... | https://mathoverflow.net/users/45564 | Multivariate CLT with varying dimension size | I interpret this as being about (Berry--Esseen-style) closeness of $X\_1 + \cdots + X\_n$ to the $d$-dimensional Gaussian distribution, when $n$ is fixed and when the dependence on the dimension $d$ is carefully taken into account.
It depends on what class of 'tests' you use to measure closeness, but the best general... | 8 | https://mathoverflow.net/users/658 | 228355 | 106,477 |
https://mathoverflow.net/questions/228303 | 6 | This question is a follow-up to my previous question [factorization of the regular representation of the symmetric group](https://mathoverflow.net/questions/227515/factorization-of-the-regular-representation-of-the-symmetric-group), which was answered in a very satisfactory way.
Let $\operatorname{Conf}(n,\mathbb{R}^... | https://mathoverflow.net/users/10273 | factorization of the cohomology of configuration space | Here's a geometric construction of a factorization that works for points in $\mathbb R^2$ (and not any other dimension). Given the close relationship between the cohomology rings of configuration spaces of points in $\mathbb R^d$ for varying $d$ (cf my answer to [Cohomology of configuration space as a representation of... | 3 | https://mathoverflow.net/users/1310 | 228358 | 106,479 |
https://mathoverflow.net/questions/228356 | 4 | A planar graph cannot have $K\_5$ and $K\_{3,3}$ as minors. Robertson-Seymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are obstructions that prevent the graph from being genus $g$. Is there any result on the size of the list? Is it linear in $g$?
| https://mathoverflow.net/users/nan | Asymptotics of list size in Robertson-Seymour theorem | No, it is not linear in the genus; it is at least exponential in $g$. See for example this [answer](https://cstheory.stackexchange.com/questions/8876/forbidden-minors-for-bounded-genus-graphs) by David Eppstein.
| 3 | https://mathoverflow.net/users/2233 | 228361 | 106,481 |
https://mathoverflow.net/questions/228283 | 1 | Could someone give a reference or construct an example of closed subspace of $Y\subset L\_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$.
I read somewhere that $Y$ is necessarily of infinite dimension and codimension.
| https://mathoverflow.net/users/19593 | Antiproximanal subspace of $L_1[0,1]$ | The $Y$ in Mikhail's answer has codimension one. Obviously $Y$ cannot be reflexive, but $Y$ can be of any non zero finite codimension or of infinite codimension. (Let $Z$ be any separable Banach space and let $Q$ be an operator from $L\_1$ to $Z$ that maps the closed unit ball of $L\_1$ onto the open unit ball of $Z$. ... | 2 | https://mathoverflow.net/users/2554 | 228362 | 106,482 |
https://mathoverflow.net/questions/228363 | -1 | The input of my problem is an integer $n\geq 3$.
The output is an integer $r\geq 1$ which must be **as small as possible** such that there is a $(n\times r)$ matrix verifying the following constraints:
* the entries of the matrix are $0$ or $1$.
* all the rows are differerent.
* in each row, there are exactly $\lfl... | https://mathoverflow.net/users/85242 | FInd smallest value $r$ such that a $n\times r$ matrix exists | There are plenty of ways to list all the $\lfloor r/2\rfloor$-subsets of $\lbrace 1,2,\ldots,r\rbrace$ such that adjacent members of the list have intersection $\lfloor r/2\rfloor-1$ (see "gray codes for subsets"). Use these as the rows of the matrix and just chop off the rows past the $n$-th.
So the answer is the mi... | 0 | https://mathoverflow.net/users/9025 | 228372 | 106,484 |
https://mathoverflow.net/questions/228370 | 2 | If the steps are iid uniform as in the title, is the return probability known? Is it positive? Answers, comments, references welcome. Clearly each of these steps is not equivalent to $d$ steps of type $\pm e\_i$, especially for large $d$.
Its distribution after $n$ steps is that of $d$ independent uniform $\pm 1$ sum... | https://mathoverflow.net/users/17773 | Does walk on $Z^d$ with steps $(\pm 1,\pm1,\ldots,\pm 1)$ return to origin? | These random walks are recurrent when $d\le 2$ and transient when $d \ge 3$. That behavior happens for a wide variety of random walks.
The expected number of returns to the origin is $$\sum\_{n=1}^\infty \left(\frac{1}{2^{2n}}{2n \choose n}\right)^d.$$
If the expected number of returns is $x \lt \infty$, then the p... | 5 | https://mathoverflow.net/users/2954 | 228373 | 106,485 |
https://mathoverflow.net/questions/228381 | 2 | For surfaces embedded in $\mathbb R^3$ with principal curvatures $ \kappa\_1, \kappa\_2 $ we know bending/isometric mappings conserve $ K= \kappa\_1 \kappa\_2 $ and CMC DeLaunay type minimal surfaces conserve $ 2 H =\kappa\_1 +\kappa\_2 $ by a physical differential pressure.
What type of deformations are conserved wi... | https://mathoverflow.net/users/47973 | Constant Harmonic Mean surfaces | Surfaces with constant $(\tfrac{1}{\kappa\_1}+\tfrac{1}{\kappa\_2})$ are contained in the class of linear Weingarten surfaces, i.e., surfaces such that
$$2aH+bK=c$$
for some $a,b,c\in\mathbb R.$
These surfaces are well-studied. They are critical points
for an energy functional which contains area, enclosed volume and... | 2 | https://mathoverflow.net/users/4572 | 228387 | 106,487 |
https://mathoverflow.net/questions/227636 | 4 | Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum\_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?
| https://mathoverflow.net/users/61426 | Mean value of a function associated with continued fractions | If $\frac{p\_{2k}}{ q\_{2k}}$ and $\frac{p\_{2k+1}}{ q\_{2k+1}}$ ($k\ge 0$) are consecutive convergents of the continued fraction expansion of $x$ then
$$\left|x-\frac{p\_{2k}}{ q\_{2k}}\right|+\left|x-\frac{p\_{2k+1}}{ q\_{2k+1}}\right|=\frac{p\_{2k+1}}{ q\_{2k+1}}-\frac{p\_{2k}}{ q\_{2k}}=\frac{1}{q\_{2k}q\_{2k+1} }.... | 4 | https://mathoverflow.net/users/5712 | 228393 | 106,489 |
https://mathoverflow.net/questions/228384 | 2 | Let $M$ be a von Neumann algebra in $B(H)$. Let $p$ and $q$
be projections in $M$. Assume that they are equivalent in $B(H)$, i.e there is a partial isometry $u$ in $B(H)$ with $p=uu^\*$ and $q=u^\*u$.
Question: Are $p$ and $q$ equivalent in $M$?
| https://mathoverflow.net/users/84390 | Equivalent projections in von Neumann algebras | The answer to your question is negative.
Take two inequivalent projections in a type $II\_1$ factor $R\subset B(H)$.
These projections have infinite dimensional range in $H$.
They are therefore equivalent in $B(H)$.
| 5 | https://mathoverflow.net/users/5690 | 228400 | 106,494 |
https://mathoverflow.net/questions/228391 | 1 | If $X$ is a normed linear space and $S(X)$ its unit sphere, $X′$ its dual space and $Π=\{(x,f)∈S(X)×S(X′) \ | \ f(x)=1\}$, then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by $V(T)=\{f(Tx) \ | \ (x,f)∈Π\}$. A projection on a complex Banach space $X$ is said to be hermitian if its numerical range i... | https://mathoverflow.net/users/34096 | Hermitian Projections on $C[0,1]$ | As I understand, the fact that for $C(0,1)$ we have too many
supporting pairs $(x,f)$ implies that there are no nontrivial
Hermitian projections in the following way. Let $t\in[0,1]$,
observe that $f=\delta\_t$ and any $x$ with $||x||=1$ and $x(t)=1$
form a supporting pair. Let $T$ be a Hermitian projection. Then
$f(Tx... | 4 | https://mathoverflow.net/users/37822 | 228404 | 106,497 |
https://mathoverflow.net/questions/228417 | 2 | Let $X$ be a set, $\tau\_1 \leq \tau\_2$ two comparable topologies on $X$ ($\tau\_1$ is weaker than $\tau\_2$) and consider some topological property $\varphi$ that holds for both $\tau\_1$ and $\tau\_2$.
I am interested in a list of properties $\varphi$ that hold for all topologies $\tau$ in between $\tau\_1$ and $\ta... | https://mathoverflow.net/users/58682 | Preservation of topological properties in between two topologies | [**Edit:** Originally my second point below stated *incorrectly* that if $\tau\_1$ and $\tau\_2$ are as described, then every topology in between is metrizable. This is not what the theorem in my paper says, but yesterday, in my haste, I thought that the theorem in my paper should imply this statement. It does not, as ... | 5 | https://mathoverflow.net/users/70618 | 228421 | 106,503 |
https://mathoverflow.net/questions/228119 | 9 | A *Frankl family* is a nonempty finite family $\mathcal F$ of nonempty finite sets such that $A,B\in\mathcal F\implies A\cup B\in\mathcal F.$ Define $d\_\mathcal F(x)=|\{A\in\mathcal F:x\in A\}|$ and $\Delta(\mathcal F)=\max\_xd\_\mathcal F(x).$ Frankl's union-closed sets conjecture says that $\Delta(\mathcal F)\gt\fra... | https://mathoverflow.net/users/43266 | A strengthening of Frankl's union-closed sets conjecture? | It seems that here is a counterexample to the stronger statement. Perhaps, it can be modified to provide one for the weaker statement?
Our universe is $\{1,\dots,7\}$. Each set of the family contains either $6$, or $7$, or both. We asume that the elements $1,\dots,5$ are arranged into a *cycle* $1-2-3-4-5-1$.
The s... | 5 | https://mathoverflow.net/users/17581 | 228428 | 106,504 |
https://mathoverflow.net/questions/228332 | 3 | I am interested in Airy's equation
$$\frac{\partial u}{\partial t}(t,x)=-\frac{\partial^3 u}{\partial x^3}(t,x)$$
on a bounded or semi-bounded domain, e.g. on $(-\infty,0)$. In order to obtain a group of isometries as propagator in $L^2(-\infty,0)$, two boundary conditions have to imposed - e.g., $u'(0)=0$ and either $... | https://mathoverflow.net/users/26039 | Airy's equation on $\mathbb R_-$ | The operators $A=-\,\partial\_x^3$ with domain $\mathscr D(A) =
\left(H^3\cap H\_0^2\right)(\mathbb R\_-)$ and $A^\ast=\partial\_x^3$
with domain $\mathscr D(A^\ast) = \left(H^3\cap H\_0^1\right)(\mathbb
R\_-)$ are $m$-dissipative on $L^2(\mathbb R\_-)$. This follows from
$\Re\langle Au,u\rangle =0$ for $u\in\mathscr D... | 3 | https://mathoverflow.net/users/69194 | 228435 | 106,505 |
https://mathoverflow.net/questions/228439 | 14 | The Martin's axiom number $\mathfrak m$ is the least cardinal $\kappa$ for which $\text{MA}\_\kappa(\text{ccc})$ is false, i.e. the least cardinal such that there exists a ccc poset $P$ and a family $\mathcal D$ of dense subsets of $P$ with $|\mathcal D| = \kappa$ such that there no $\mathcal D$-generic filter $G \subs... | https://mathoverflow.net/users/76006 | Is the Martin's axiom number $\mathfrak m$ regular | Not necessarily. That $\mathfrak m$ is consistently singular is proved in
>
> [MR0947850 (89m:03045)](http://www.ams.org/mathscinet-getitem?mr=947850) Kunen, Kenneth. [*Where $\mathsf{MA}$ first
> fails*](http://www.jstor.org/stable/pdf/2274515.pdf). J. Symbolic Logic **53(2)**, (1988), 429–433.
>
>
>
There,... | 15 | https://mathoverflow.net/users/6085 | 228440 | 106,507 |
https://mathoverflow.net/questions/228445 | 11 | Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of
permutations on an $N$-element set that have exactly $m$ cycles (counting
$1$-cycles). Then it is in the literature that the polynomial generating
function
$$
\sum\_{m=0}^N C(m)x^m = x(x+1) \dots (x+N-1),
$$
the rising factorial.
I would like a r... | https://mathoverflow.net/users/42278 | Characters of permutation groups | This follows from using generating functions a la Polya. By the cycle index formula, one sees that (using the notation of $C\_n(k,j)$ for permutations in $S\_n$ with $k$ cycles including $j$ fixed points)
$$
\sum\_{n=0}^{\infty} \frac{t^n}{n!} \sum\_{k,j} C\_n(k,j) y^j x^k
= \exp\Big( tx y +\sum\_{j=2}^{\infty} \frac... | 7 | https://mathoverflow.net/users/38624 | 228448 | 106,508 |
https://mathoverflow.net/questions/228458 | 3 | Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow \tau(S^m)\longrightarrow S^m.
$$
The Serre spectral sequence has $E\_2$-page
$$
E\_2^{p,q}=H^p(S^{m};\mathbb{Z})\otimes H^q(S^{m-1};\mathbb{Z... | https://mathoverflow.net/users/65800 | cohomology module of unit tangent vector bundles over spheres | If $m$ is even, $d\_m$ is multiplication by $2$:
For every unit sphere bundle of an $m$-dimendional vector bundle over a space $X$, $d\_m$ sends $[S^{m-1}] \otimes 1$ to the euler class $e(x) \in H^m(X)$. In your case, we know that $$\langle e, [S^m]\rangle = \chi(S^m) = 1+(-1)^m$$ which is $2$ if $m$ is even. You c... | 4 | https://mathoverflow.net/users/14233 | 228465 | 106,515 |
https://mathoverflow.net/questions/228467 | 2 | Let $M$ be a W\*-algebra. I am looking for the proof of the following fact:
Let $z$ be the supremum of minimal projections in $M$. Then $z$ is central.
| https://mathoverflow.net/users/84390 | Minimal central projection in W*-algebras | It is enough to show that $z$ commutes to all unitaries. But if $p$ is a minimal projection then $upu^{-1}$ is again a minimal projection.
| 4 | https://mathoverflow.net/users/22131 | 228472 | 106,518 |
https://mathoverflow.net/questions/228451 | 4 | I have seen this question asked at least once before, but not with any real answers.
I was reading about the various reconstruction conjectures and equivalents, and I saw that the reconstruction conjecture, which uses a multi-set, has a set counterpart, the set reconstruction conjecture.
My question is: Does the ed... | https://mathoverflow.net/users/85294 | Edge Reconstruction Conjecture | The edge-reconstruction conjecture does have an analogous statement to the set version of the reconstruction conjecture. As far as I am aware there is no graphs that are edge-reconstructible but not "set edge-reconstructible". Therefore the corresponding conjecture would (probably) state that every graph with at least ... | 3 | https://mathoverflow.net/users/69775 | 228475 | 106,519 |
https://mathoverflow.net/questions/228453 | 1 | **Definition** Doubling dimension ($\dim\_D(M)=k$): A Metric space $M=(V,d)$ has doubling dimension at most $k$ if for any $x\in V$ & $r>0$, $B(x,r)\subseteq\bigcup^{2^k}\_{i=1}B(x\_i,r/2)$.
With this definition, a paper that i have read stated without proof $\dim\_D(N)\leq2\dim\_D(V)$ for any net $N\subseteq V$, whe... | https://mathoverflow.net/users/80289 | Why is the doubling dimension of any net of a metric space at most half of that of the metric space? | This holds for *every* subset $N\subseteq V$, not only for nets.
Every ball $B(x,r)$ is covered by a union of $2^{2k}$ balls $B(x\_i,r/4)$. Now, if for some $i$ there exists $n\_i\in B(x\_i,r/4)\cap N$, then $B(x\_i,r/4)\subseteq B(n\_i,r/2)$. So such balls $B(n\_i,r/2)$ cover $B(x,r)\cap N$.
| 0 | https://mathoverflow.net/users/17581 | 228477 | 106,520 |
https://mathoverflow.net/questions/226732 | 20 | The $n$-th harmonic number is defined as
$$
H\_n=\sum\_{k=1}^{n}\frac{1}{k},
$$
and the generalized harmonic numbers are defined by
$$
H\_{n}^{(r)}=\sum\_{k=1}^{n}\frac{1}{k^r}.
$$
Recently, I have found the following combinatorial identity involving the second-order harmonic numbers (I have computational evidence).
... | https://mathoverflow.net/users/6104 | A combinatorial identity involving generalized harmonic numbers | The identity
$$
\begin{align}
\sum\_{s=1}^{m}{2s\choose s}{s\choose m-s}\frac{(-1)^s }{s+1}H\_{s}^{(2)}=\frac{2(-1)^m}{m+1}\sum\_{s=1}^m H\_{s}^{(2)}. \tag{1}
\end{align}
$$
is equivalent to the following identity
$$
\sum\_{s=1}^{m}{2s\choose s}\frac{H\_s^{(2)}}{s+1}(x-x^2)^s=\frac{2\text{Li}\_2(x)}{1-x}-\frac{\ln^2(1-... | 14 | https://mathoverflow.net/users/82588 | 228483 | 106,522 |
https://mathoverflow.net/questions/228468 | 3 | In Bishop's constructive mathematics, is there any literature on a possible version of the weak König's lemma, or of the compactness theorem for countable models? There is some related information [here](https://mathoverflow.net/questions/136371) but not enough to resolve the issue.
| https://mathoverflow.net/users/28128 | Compactness in Bishop's constructive mathematics | A detailed analysis of König's lemma in Bishop-style constructivism was carried out by Hajime Ishihara, Josef Berger and Helmut Schwichtenberg. Some references:
1. *Hajime Ishihara*, [**Weak König’s lemma implies Brouwer’s fan theorem: a direct proof**](http://dx.doi.org/10.1305/ndjfl/1153858649), *Notre Dame J. Form... | 8 | https://mathoverflow.net/users/1176 | 228490 | 106,525 |
https://mathoverflow.net/questions/228489 | 12 | I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article *Inequalities for numerical invariants of sets of matrices*. Let $\|\cdot\|$ denote the matrix norm induced by the Euclidean norm, and let $\rho(A)$ denote the spec... | https://mathoverflow.net/users/1840 | An inequality for the spectral radius of matrices used by J. Bochi | Constant is definitely not sharp. For example, let me show inequality $\|A^d\|\leqslant \frac1{\sqrt[d]{2}-1}\cdot \rho(A)\cdot \|A\|^{d-1}$. We may suppose $\|A\|=1$, denote $\rho(A)=\rho$, then we have two inequalities $\|A^d\|\leqslant \|A\|^d=1$ and $\|A^d\|\leqslant d\rho+\dots+\rho^d=(1+\rho)^d-1$, explained in t... | 14 | https://mathoverflow.net/users/4312 | 228493 | 106,526 |
https://mathoverflow.net/questions/228464 | 0 | Let $x$ be a binary random variable and $z$ be an arbitrary random variable. $x$ and $z$ are, in general, not independent.
Let $y\_1, \ldots y\_n$ be $n$ identically distributed binary random variables conditionally independent given $z$.
A graphical model would have $x$ at the root, pointing to $z$, and $z$ pointi... | https://mathoverflow.net/users/8737 | Local extrema of a posterior probability | I think you can get as many as you like. Let X be 0 or 1 with probability 1/2 each. Let $Z|X=I$ be of the form $h \pm \epsilon q$ where q is arbitrary and can be positive or negative, and I write them as if they have a density but any measure/signed measure will, subject to the sum being a measure. First I claim that
... | 1 | https://mathoverflow.net/users/nan | 228501 | 106,529 |
https://mathoverflow.net/questions/228418 | 5 | Here we are considering subsets $\mathcal{F}$ of $2^\omega$, which are in correspondence with families of subsets of $\omega$ (sets of "reals"). Such a family is *Borel* if it is a Borel subset of $2^\omega$ under the usual topology.
Such a family is *almost disjoint* if, for every pair $X\not=Y$ from $\mathcal{F}$, ... | https://mathoverflow.net/users/15735 | Is there an uncountable Borel almost disjoint family? | Firstly, as commented by Asaf Karagila, it is routine to build a Borel uncountable family of almost disjoint subsets of $\omega$: fix a one-to-one map $f$ from the set $2^{<\omega}$ of finite binary sequences to $\omega$. It is clear that if $B$ and $B'$ are any two distinct branches of $2^{<\omega}$, then $f(B)$ and $... | 16 | https://mathoverflow.net/users/9269 | 228506 | 106,531 |
https://mathoverflow.net/questions/228496 | 3 | Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
1. If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" matrix satisfying: $A\times adj^{Pf}(A)=Pf(A)Id$. What is the standard notation for this $adj^{Pf}(A)$? (And the stan... | https://mathoverflow.net/users/2900 | the pfaffian-adjugate and its counterparts for matrices odd size | These questions are all answered in terms of the exterior algebra over the ring $R$ of the free module $M=R^n$. Namely, the skew-symmetric matrices naturally live in $\Lambda^2(M) \simeq R^N$ where $N=\tfrac12n(n{-}1)$. The top exterior power is $\Lambda^n(M)\simeq R$.
If $n=2m$, and $A\in\Lambda^2(M)$ is given, the... | 5 | https://mathoverflow.net/users/13972 | 228509 | 106,534 |
https://mathoverflow.net/questions/211343 | 1 | Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector,
and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v\_0, ... , v\_n$, where $n\ge m$).
Let's further assume that
* $dist(v\_i,H) = 0,\ i\in [0,k]\ \wedge\ dist(v\_i,H) \gt 0,\ i\in[k+1,n]$
* $dist(v\_j,H') = 0,\ j... | https://mathoverflow.net/users/31310 | An optimality condition for the corners of convex polytopes? | No. Take a triangle $v\_0v\_1v\_2$ on the plane, $H'$ is its side $v\_0v\_1$, $H$ is almost another side $v\_0v\_2$. Then $\sum\_{i=0}^2 {\rm dist}\, (v\_i,H')$ is just a length of altitude from $v\_2$, $\sum\_{i=0}^2 {\rm dist}\, (v\_i,H)$ is almost the length of altitude from $v\_1$, which may be less than that from ... | 1 | https://mathoverflow.net/users/4312 | 228511 | 106,535 |
https://mathoverflow.net/questions/227888 | 5 | Let $G$ be an undirected graph.
* The eccentricity of a vertex $v$ of $G$, is the maximum distance between $v$ and any other vertex of $G$:$\;\;$ $\mathit{ecc}(v) = \max\_{u}\mathit{dist}(v,u)$.
* The radius of $G$ is the minimum eccentricity of a vertex of $G$:$\;\;$$R(G) = \min\_v \mathit{ecc}(v)$.
* The diameter o... | https://mathoverflow.net/users/82650 | Diameter vs Radius in Maximal Planar Graphs | The inequality
$$R(G)\leq \lfloor\frac{D(G)}{2}\rfloor + 1$$
does not hold in general for maximal planar graphs. For a counterexample, let $G$ be the icosahedral graph. Every face of the icosahedron is a triangle, and a planar graph is maximal if and only if it is a triangulation, so $G$ is a maximal planar graph. The ... | 3 | https://mathoverflow.net/users/22055 | 228517 | 106,538 |
https://mathoverflow.net/questions/223260 | 11 | Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of $\sigma$ such that $\Gamma(\sigma)$ isn't a pole, this will appear to be (almost) a two-sided probability density functio... | https://mathoverflow.net/users/24611 | Probability distribution derived from gamma function - does it have a name? | This is called Generalized Hyperbolic Secant Distributions. See Generalized Hyperbolic Secant Distributions of W. L. Harkness and M. L. Harkness (<http://www.jstor.org/stable/2283852>).
In particular, see equation (3) in that paper and use the fact that $\overline{\Gamma(z)}=\Gamma(\bar z)$
| 5 | https://mathoverflow.net/users/85303 | 228518 | 106,539 |
https://mathoverflow.net/questions/228534 | 3 | What would the best resources be for someone who wants to study self-avoiding walks from a mathematical standpoint?
I'm talking about seminal/important papers, good textbooks perhaps, things of that nature.
| https://mathoverflow.net/users/85334 | Resources to study self-avoiding walks | There are recent lecture notes by Bauerschmidt, Duminil-Copin, Goodman, and Slade (arXiv:1206.2092). Most of the important papers should be in the references to that lecture notes, and it depends on your focus which of those to recommend. For the 2D case, you may look at
G.F. Lawler, O. Schramm, and W. Werner, On the... | 4 | https://mathoverflow.net/users/56624 | 228545 | 106,545 |
https://mathoverflow.net/questions/228525 | 5 | Recently, I saw a question in [see here](http://www.puzzling.stackexchange.com) which is so interesting for me. This question is as follows:
>
> Is it possible to fill the $121$ entries in an $11×11$ square with the values $0,+1,−1$, so that the row sums and column sums are $22$ distinct numbers?
>
>
>
This pr... | https://mathoverflow.net/users/84430 | reverse definition for magic square | This problem is so famous. For first trivial reference, you can see:[link](http://mathworld.wolfram.com/-101-Matrix.html).
$\it{R. Bodendiek}$ and $\it{G. Burosch}$ studied this problem in a paper with name:
"Solution to the Antimagic 0,1,-1 Matrix Problem."
If there is solution for integer $n$, then we have:
$... | 4 | https://mathoverflow.net/users/19885 | 228554 | 106,546 |
https://mathoverflow.net/questions/228560 | 5 | Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying to understand what properties $X$ needs to have so that the probability of the Brownian particle striking $X$ within tim... | https://mathoverflow.net/users/85355 | Brownian motion in $\mathbb{R}^n$, probability of hitting a set | It's not that simple. See about polar/nonpolar points/sets e.g. in <http://wiki.math.toronto.edu/TorontoMathWiki/index.php/Brownian_Motion_and_Harmonic_functions>
If I remember correctly, a set is not polar iff it has positive capacity (w.r.t. logarithmic potential in two dimensions, and Newton potential for $d\geq 3... | 5 | https://mathoverflow.net/users/81488 | 228562 | 106,549 |
https://mathoverflow.net/questions/166047 | 5 |
>
> Let $U\_n=\sum\_{i=1}^n X\_i,V\_n=\sum\_{i=1}^n Y\_i$, $n\geq 1$, be a two-dimensional random walk with i.i.d. increments $(X\_n, Y\_n)$, where $X\_n, Y\_n$ are discrete random variables with joint pmf $P\_{X,Y}$. $X\_n,Y\_n$ have the following properties:
> \begin{align}
> 0 < \mathbb{E}[X\_n]=\mu\_X, \quad 0
>... | https://mathoverflow.net/users/50758 | Stopping time of two dimensional random walk | Note that
$$\mathbf E[\tau(t)]=\sum\_{j=0}^\infty \mathbf P(\tau(t)>j)\le t+1+\sqrt{t}+\sum\_{j=[t+\sqrt{t}]+1}^\infty \mathbf P(\tau(t)>j).$$
To bound the sum we use $\{\tau(t)>j\}\subset \{U\_j<t\mu\_X\mbox{ or } V\_j<t\mu\_Y\} $. Therefore,
$$
\mathbf P(\tau(t)>j)\le \mathbf P(U\_j<t\mu\_X)+\mathbf P(V\_j<t\mu\_Y... | 3 | https://mathoverflow.net/users/85303 | 228564 | 106,550 |
https://mathoverflow.net/questions/228432 | 5 | What is known about the algebraic variety $V\_G$ defined by $det(X\_G) = 1$ where $X\_G$ is the group matrix $(x\_{g\_ig\_j^-1})$ of a finite group $G$? It is known that two finite groups having the same determinant are isomorphic.
**Edit:** Are there any results concernig the group structure on $V\_G$?
| https://mathoverflow.net/users/nan | What is known about the algebraic variety defined by the group determinant? | Per the suggestion of Neil Strickland, here is an answer. There are some excellent [notes](http://www.math.uconn.edu/~kconrad/articles/groupdet.pdf) of Keith Conrad about all of this.
If $k$ is an algebraically closed field of characteristic prime to $n$, the order of $G$, then classical results about representations... | 4 | https://mathoverflow.net/users/13265 | 228569 | 106,554 |
https://mathoverflow.net/questions/228575 | 3 | I have been troubled by this seemingly simple question recently.
How do we easily visualize the statement:
>
> Surgery of $S^3$ over a trivial unknot gives $S^1 \times S^2$?
>
>
>
All I can think of for the first step is to remove a point from $S^3$ so we can work on $R^3$, but then I have no idea how to go o... | https://mathoverflow.net/users/85361 | Surgery of $S^3$ | $S^3$ is a union of two solid tori, glued along their boundary, where a meridian is glued to the longtitude, and conversely. When you do the surgery, you get a solid torus doubled along the boundary. If you think of a solid torus as $D^2 \times S^1,$ the double is a double of $D^2$ times $S^1.$ The double of $D^2$ is $... | 4 | https://mathoverflow.net/users/11142 | 228576 | 106,556 |
https://mathoverflow.net/questions/228581 | 5 | My question is a little bit vague. I want to know if an arbitrary compact Riemannian manifold (M^d,g) admits partitions that are uniform in some sense. To be more precise, I need for every eps > 0 a partition of M into finitely many measurable subsets P\_1,...,P\_k such that
(i) The diameter of each P\_i is smaller ... | https://mathoverflow.net/users/nan | Uniform partitions of a compact Riemannian manifold | Assume first that your manifold $M$ is embedded in an Euclidean space $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$ and the metric is the induced metric. (Nash's embedding theorem shows that this is always possible.)
For each $\newcommand{\ve}{\varepsilon}$ $\ve>0$ denote by $\newcommand{\eL}{\mathscr{L}}$ $\eL\_{\ve}$ the ... | 3 | https://mathoverflow.net/users/20302 | 228583 | 106,561 |
https://mathoverflow.net/questions/228199 | 0 | suppose $f:X\rightarrow Y$ is a morphism between two schemes over scheme $S.$ Do we have the morphism between their hilbert schemes, i.e. is there a natural morphism $Hilb(X/S)\rightarrow Hilb(Y/S)$ over $S$? What about the Douady spaces instead of Hilbert schemes in the analytic spaces case?
| https://mathoverflow.net/users/42804 | functoriality of hilbert scheme | I am just posting my comments above as an answer, so that this question does not remain unanswered.
Yes, that is also true for the Douady spaces. This has nothing to do with the construction or proof of existence of the Hilbert schemes / Douady spaces. Already for the (contravariant) Hilbert functor $$ \underline{\t... | 2 | https://mathoverflow.net/users/13265 | 228589 | 106,563 |
https://mathoverflow.net/questions/228473 | 9 | A particular case of Lurie and Hopkins' ambidexterity theory is that if $G$ is a finite group acting on a $K(n)$-local spectrum $X$ then the norm map
$$ X\_{hG} \to X^{hG} $$
is a $K(n)$-local equivalence. If we now replace $G$ with a compact lie group and $X$ with a genuine equivariant **free** $G$-spectrum (not neces... | https://mathoverflow.net/users/51164 | Genuine equivariant ambidexterity | This is to address Yonatan's question in the comments. Let $G$ be a finite group. To every (genuine) $G$-spectrum $E$, you can associate its (genuine) fixed point spectrum $E^{G}$. This earns its name by virtue of the following compatibility between stable and unstable homotopy theory:
$\Omega^{\infty}( E^G ) = (\Omega... | 8 | https://mathoverflow.net/users/7721 | 228593 | 106,566 |
https://mathoverflow.net/questions/228497 | 8 | Katz defines in Section 2.0 [$p$-adic properties of modular schemes and modular forms](https://web.math.princeton.edu/~nmk/old/padicpropMFMS.pdf) the Hasse invariant as a mod $p$ modular form $A$ of weight $p-1$. In other words, it is a section of $\omega^{\otimes p-1}$ on the compactified moduli stack of elliptic curv... | https://mathoverflow.net/users/2039 | Lifting the Hasse invariant mod $2$ | $\newcommand\Q{\mathbf{Q}}$
$\newcommand\Z{\mathbf{Z}}$
$\newcommand\Zbar{\overline{\Z}}$
$\newcommand\F{\mathbf{F}}$
$\newcommand\Gal{\mathrm{Gal}}$
A lift always exists, even with coefficients in $\Z$. If suffices to consider the case when $p > 2$ is prime. (**Added Note**: there is a second elementary argument at ... | 7 | https://mathoverflow.net/users/85372 | 228596 | 106,568 |
https://mathoverflow.net/questions/228578 | 18 | I read [this](http://www.ams.org/journals/notices/201601/rnoti-p23.pdf) interview with Ian Agol, where he says:
>
> "...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots."
>
>
>
My question is:
>
> How does this work exactly? Is this written up somewh... | https://mathoverflow.net/users/84120 | Classification of knots by geometrization theorem | You have all the tools to compute the geometric decomposition of knot and link exteriors in the software [Regina](http://regina.sourceforge.net/). I'm one of the authors, although my hands haven't been over that part of the code very much. Most of the 3-manifold decomposition code was written by Ben Burton. The algorit... | 14 | https://mathoverflow.net/users/1465 | 228616 | 106,580 |
https://mathoverflow.net/questions/228533 | 3 | 1. Every connected graph has a spanning tree.
2. Every non-empty set can be well ordered.
Basically I am trying to show that statement 1 implies statement 2. What I tried is as following:
Let $X \ne \emptyset$, define a graph $G$ with $$V(G) := \{v\_{S}:S \subseteq X\}$$ and $v\_S v\_T \in E(G)$ if and only if either... | https://mathoverflow.net/users/nan | Existence of Spanning Tree implies Well Ordering Principle | Let AC denote the axiom of choice. The proof of the implication AC $\implies$ (2) is somewhat nontrivial. Inasmuch as the equivalence AC $\iff$ (1) is quite trivial, it seems unlikely that any "direct" proof of the implication (1) $\implies$ (2) will be much simpler than (1) $\implies$ AC $\implies$ (2). The rest of th... | 8 | https://mathoverflow.net/users/43266 | 228617 | 106,581 |
https://mathoverflow.net/questions/228584 | 3 | You play the following game.
1. You get $4n$ gold coins and have to arrange them in the unit square in general position (no two coins have the same x or the same y coordinate). Call this set of coins $S\_1$.
2. There is a lottery in which each coin in $S\_1$ is selected with probability 1/2, independently of the othe... | https://mathoverflow.net/users/34461 | A lottery on coins in a convex set | One quick upper bound for the axis-parallel rectangle case for large $n$: Fix an arbitrary arrangement of $4n$ points. Then there are at most $cn^4$ distinct subsets of $2n$ points that lie inside axis-parallel rectangles containing exactly $2n$ points (the points in a rectangle are determined by an uppermost, lowermos... | 3 | https://mathoverflow.net/users/405 | 228623 | 106,583 |
https://mathoverflow.net/questions/122454 | 17 | I have an $n$ step random walk which starts at zero $X\_0 = 0 = S\_0$ where the steps $X\_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it stays in the right half-line $[0,\infty)$ (that is, $S\_k = X\_0 + X\_1 + \dots + X\_k \geq 0$ for $k \in 0, \dots, n$).... | https://mathoverflow.net/users/9101 | Distribution of maximum of random walk conditioned to stay positive | 1) I believe it is possible to obtain $P(M\_n<a\ | \tau>n)$, where $\tau:=\inf\{n\ge 1: S\_n\le 0\}$ in more or less explicit way only for the case of the simple random walk. These computations can be found in Billingsley, Convergence of Probability measures, Chapter 2.11. For large $n$ these computations become univer... | 3 | https://mathoverflow.net/users/85303 | 228626 | 106,585 |
https://mathoverflow.net/questions/228624 | 28 | I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas.
The audience is familiar with characteristic classes at the level of Milnor Stasheff. We are no experts on homotopy theory.
What would you recommend?
Edit: ... | https://mathoverflow.net/users/12156 | Book recommendation for cobordism theory | Perhaps the [Notes on cobordism](http://www-math.mit.edu/~hrm/papers/cobordism.pdf) by Haynes Miller could be of some help too.
Another possibility (but geared primarily towards applications in symplectic geometry) is the book
V. Guillemin, V. Ginzburg, Y. Karshon, Moment maps, cobordisms, and Hamiltonian group act... | 15 | https://mathoverflow.net/users/85395 | 228639 | 106,591 |
https://mathoverflow.net/questions/228574 | 18 | Maybe this question has already been considered here, but after a quick search I didn't find what I was looking for.
As I see, in the literature there are two different definitions of the topological/Lebegue covering dimension.
$\textbf{Definition 1.}$ (e.g. Munkres, General topology) A topological space $X$ is sai... | https://mathoverflow.net/users/85244 | Two definitions of Lebesgue covering dimension | As you refer to Engelking's "Dimension theory" book, I suppose you know the following two statements, but anyway, here they are:
* The notions agree for separable metric spaces, by Exercise 1.7.E of Engelking's book.
* The notions agree for paracompact spaces by Proposition 3.2.2 of Engelking's book. (In particular, ... | 20 | https://mathoverflow.net/users/50846 | 228642 | 106,593 |
https://mathoverflow.net/questions/228653 | 2 | Consider a graph with $n$ vertices such that if one takes any 4 vertices there are at most 4 edges among these 4 vertices (Notice that there are 6 "possible" edges among these 4 vertices). What is the maximum possible edge density for such a graph as a function of $n$ and what is the limit as $n$ goes to infinity?
Us... | https://mathoverflow.net/users/85402 | Minimum Edge Density given a particular condition | It is well-known, but I do not know where did it appear for the first time.
The answer is $f(2k)=k^2$ for $n=2k$ and $f(2k+1)=k(k+1)$ for $n=2k+1$. Examples are bipartite graphs $K\_{k,k}$ and $K\_{k,k+1}$. The proof of upper estimate goes by induction. Base $n=4$ is clear. Assume that $n\geqslant 5$ and graph on $n-... | 2 | https://mathoverflow.net/users/4312 | 228656 | 106,597 |
https://mathoverflow.net/questions/219585 | 1 | Is there an operator $T:X\rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l\_{1}$, but does not factor through any Banach space $W$ containg no copy of $l\_{1}$?
| https://mathoverflow.net/users/41619 | An operator factoring through a Banach space containing no copy of $l_{1}$ | This is an explanation of Johnson's answer. We need to prove two things: (1) An identity on $C(0,1)$ does not factor through space which does not contain $\ell\_1$. (2) $C(0,1)$ does not contain a complemented copy of $\ell\_1$ (of course the identity on $C(0,1)$ factors through $C(0,1)$).
(1) By the Banach-Mazur th... | 1 | https://mathoverflow.net/users/85406 | 228663 | 106,600 |
https://mathoverflow.net/questions/228628 | 2 | Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states that one can describe a complex manifold $\mathcal{M}$, which as a set is in bijection with $M$.
My question is: The ... | https://mathoverflow.net/users/2095 | Moduli of stable bundles - analytic approach | $\mathcal{M}$ is what is called a coarse moduli space. In concrete terms, this means the following:
1) As a set, $\mathcal{M}$ can be viewed as the set of isomorphism classes of stable bundles (of rank $r$ and degree $d$):
2) Given any family of such vector bundles parametrized by an analytic space $S$ (that is, a ... | 6 | https://mathoverflow.net/users/40297 | 228664 | 106,601 |
https://mathoverflow.net/questions/228661 | 11 | The following question was something that came to my mind during my (unsuccessful) attempt at answering [this MO-question.](https://mathoverflow.net/questions/228568/)
Let $X$ be a topological space, and let $\tilde{X}\to X$ be a CW-approximation. Given that $X$ has covering dimension $n$, can anything be said about ... | https://mathoverflow.net/users/50846 | Dimension in CW-approximation | Barratt and Milnor (*An Example of Anomalous Singular Homology*) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending to $0$ is non-trivial in arbitrarily high dimensions. Thus any CW-replacement of this space is infinite-dimensional. On t... | 22 | https://mathoverflow.net/users/12547 | 228666 | 106,602 |
https://mathoverflow.net/questions/220586 | 6 | We consider a discretization of the Laplace operator on $\mathbb Z^2$, <https://en.wikipedia.org/wiki/Discrete_Laplace_operator>
Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)|\leq C \ln|x|,\Delta u = 1$ at $(0,0)$ and $\Delta u=0$ elsewhere. I am sure that somewhere it is proven that the d... | https://mathoverflow.net/users/4298 | Fundamental solution of Discrete Laplace in the plane | For nearest neighbor Laplacian, you can find a short self-contained proof of the formula $$u(x)=\frac1{2\pi}\log |x|+c+O\left(\frac1{|x|^2}\right)$$ at [these lecture notes](https://wiki.helsinki.fi/download/attachments/155233241/LectureNotes20_04.pdf?version=2&modificationDate=1430322536394&api=v2), pages 6-7. It is a... | 6 | https://mathoverflow.net/users/56624 | 228673 | 106,603 |
https://mathoverflow.net/questions/228671 | 16 | I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none *seemed* to address it, but I could just be unfamiliar with the language; so please pardon me if this is a question with a well known answer.
If it makes... | https://mathoverflow.net/users/2383 | Witt-vector vectors | It is impossible if you want reasonable functoriality, and for finite-dimensional vector spaces to be carried to finite free modules for $A$ a field.
Presumably for $A = k$ a finite field of characteristic $p$, you want the composition of such a functor with "reduction modulo $p$" to be (canonically isomorphic to) th... | 14 | https://mathoverflow.net/users/81332 | 228675 | 106,604 |
https://mathoverflow.net/questions/228676 | 0 | **Question--quick version:** Does there exist a (nonconstant) polynomial $f \in \mathbb C[x,y,z]$ such that for all $c \in \mathbb C$, the affine hypersurface cut out by $f + c$ is singular?
**Motivated version:** Suppose you have a singular hypersurface $\mathbf V(f)$ in $\mathbb C^3$. Since a generic hypersurface i... | https://mathoverflow.net/users/5094 | Pencil of singular affine hypersurfaces differing only in the constant term | As you observe, there is no base locus on the affine part. This allows you to use the Bertini theorem as stated in Hartshorne Corollary III.10.9 and Remark III.10.9.2. The latter says that the projectivity assumption can be dropped if the linear system is finite-dimensional, which is certainly the case in your question... | 3 | https://mathoverflow.net/users/82179 | 228677 | 106,605 |
https://mathoverflow.net/questions/228679 | 6 | Factorization structures have been popular in the past decade. Recently a variant of this structure has been suggested by Ivan Mirkovic (and possibly collaborators). This variant, which goes under the name of "local space" is supposetly useful in understanding the affine Grassmanian and geometric Langlands.
Could a ... | https://mathoverflow.net/users/41301 | What are local spaces and what are they good for? | Local spaces:finite subschemes::Factorization spaces:finite subsets.
A local space over X is a compatible collection of spaces over the Hilbert schemes of arbitrary numbers of points in X satisfying a factorization property for disjoint union. This is very close to the notion of a factorization space, in which we hav... | 12 | https://mathoverflow.net/users/582 | 228680 | 106,606 |
https://mathoverflow.net/questions/227162 | 0 | Notations:
$R$- Noetherian graded ring and $I,J$ homogeneous ideals in $R$
**Definition:**
The projective dimension of $R/I$, denoted $pd(R/I)$, is the length of a minimal free graded resolution of $R/I$:
$0 \rightarrow \bigoplus\_{j} R(-j)^ {\beta\_{p,j}(R/I)}\rightarrow \bigoplus\_{j} R(-j)^ {\beta\_{p-1,j}(R/I)}... | https://mathoverflow.net/users/68302 | Properties of Betti number of ideal | I believe the answer to your question is no, we cannot conclude that $\beta\_{(i+j+2), 2(i+j+2)}(IJ)\neq0$.
For example, let us take $n=m=4$ and $I$ and $J$ to be the following ideals
$$I = \langle x\_1x\_2, x\_2x\_3\rangle \subset \mathbb{Q}[x\_1, x\_2, x\_3, x\_4]=R$$
$$J = \langle y\_1y\_2, y\_3y\_4\rangle \subset... | 3 | https://mathoverflow.net/users/23444 | 228681 | 106,607 |
https://mathoverflow.net/questions/228682 | 0 | We have $A\_i , B\_i , C\_i , D\_i , E\_i ,F\_i, \ (i=1, 2) $.
We want to find $ (u,v) \in \mathbb{R}^2$ satisfying
\begin{equation}
A\_1 u^2 + B\_1 uv + C\_1 v^2 + D\_1 u + E\_1 v +F\_1 =0 \\
A\_2 u^2 + B\_2 uv + C\_2 v^2 + D\_2 u + E\_2 v +F\_2 =0.
\end{equation}
Is there a general method to find exact solution... | https://mathoverflow.net/users/78387 | Cross section point of two conics curves | Take the resultant of the two equations with respect to one of the variables, say $v$. You'll get a quartic polynomial in $u$ whose coefficients are (complicated) polynomials in $A\_1,\ldots,F\_2$, but if $A\_1,\ldots,F\_2$ are real numbers, as you seem to suggest, then you'll get a quartic in $u$ with real coefficient... | 2 | https://mathoverflow.net/users/11926 | 228686 | 106,611 |
https://mathoverflow.net/questions/228678 | 3 | Let $x\_0,\dots,x\_n$ be a collection of variable points in $\mathbb{R}^2$ and let $c>0$ be a fixed constant. Is there any way I could compute an upper bound of the volume of the region in $\mathbb{R}^{2(n+1)}$ consisting of all points where $$\|x\_0\|+\sum\_{i=0}^{n-1}\|x\_i-x\_{i+1}\|\leq c$$and $$\|x\_i\|\leq 1$$ fo... | https://mathoverflow.net/users/85411 | The volume of a region arising from planar linkages | Let $V$ be the volume of the configuration space that you describe in the question. I'll record here the two trivial upper bounds for $V$ arising from dropping either the second set of inequalities $\|x\_i\|\leq 1$ or from dropping the first inequality. For $c$ sufficiently large or small, these bounds become tight, as... | 5 | https://mathoverflow.net/users/353 | 228702 | 106,617 |
https://mathoverflow.net/questions/228689 | 8 | Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
| https://mathoverflow.net/users/19075 | The parity of the full automorphism group order of finite non-abelian groups of prime exponent | Yes:
>
> for every prime $p\ge 7$, there's a finite group of exponent $p$ whose automorphism group is a $p$-group.
>
>
>
**Initial answer** *(Jan 18' 2016)*
Start from any (finite-dimensional) complex nilpotent Lie algebra $\mathfrak{g}$ that is defined over $\mathbf{Q}$ and has a unipotent automorphism group ... | 13 | https://mathoverflow.net/users/14094 | 228707 | 106,618 |
https://mathoverflow.net/questions/187948 | 1 | Let $X$ be an affine building. Assume that $X$ is periodic, by which I mean that there exists a covering $X\to F$ of a finite simplicial complex. Let $\Gamma$ denote the group of deck transformations, then $F=\Gamma\backslash X$. Call an apartment $A$ periodic, if $\Gamma\_A\backslash A$ is compact, where $\Gamma\_A$ i... | https://mathoverflow.net/users/nan | Question on affine buildings | This is exactly Theorem 8.9 from this paper :
*Werner Ballmann and Michael Brin*, [**Orbihedra of nonpositive curvature**](http://www.numdam.org/item?id=PMIHES_1995__82__169_0), *Inst. Hautes \'Etudes Sci. Publ. Math.* (1995), no. 82, 169--209 (1996).
| 1 | https://mathoverflow.net/users/81562 | 228708 | 106,619 |
https://mathoverflow.net/questions/228706 | 11 | Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as the 3-dimensional Heisenberg group), endowed with a left-invariant Riemannian metric.
The following statement can be... | https://mathoverflow.net/users/6206 | Quasi-isometric rigidity of Nil | Here's a sketch of proof (which intersects yours):
Let $\Gamma$ be QI to NIL. As you say, by Gromov's theorem, $\Gamma$ is virtually nilpotent; let some finite index subgroup be a lattice in some simply connected nilpotent Lie group $G$.
The growth of $G$ and NIL are equivalent, hence the growth of $G$ is $\simeq r... | 11 | https://mathoverflow.net/users/14094 | 228717 | 106,621 |
https://mathoverflow.net/questions/228718 | 7 | Studying integration over unitary group I came across this function, the Weingarten function Wg, such that
$$ \int\_{\mathcal{U}(N)} \prod\_{k=1}^{n} U\_{i\_kj\_k}
U^\*\_{m\_k r\_k} dU=\sum\_{\tau,\sigma\in S\_n} {\rm
Wg}^U(\tau^{-1}\sigma)\prod\_{k=1}^n \delta\_{i\_k,\tau(m\_k)}\delta\_{j\_k,\sigma(r\_k)}.$$ I underst... | https://mathoverflow.net/users/83671 | Weingarten function for unitary group | I would recommend [Elementary derivation of Weingarten functions of classical Lie groups](http://arxiv.org/abs/1406.2182) by Marcel Novaes (2015).
>
> Previous works where Weingarten functions were obtained were based
> either on representation theory and Schur-Weyl duality, the theory of
> Gelfand pairs, or Jucy... | 6 | https://mathoverflow.net/users/11260 | 228721 | 106,623 |
https://mathoverflow.net/questions/228712 | 1 | Hello I have started working on my PhD a short while ago and wondered if there might be any good introductions to spatial dynamics.
I have a basic understanding of dynamical systems but would like to expand further on it.
Thank you for your recommendations.
| https://mathoverflow.net/users/85428 | What are good references for spatial dynamics? | Based on your comment, the book
M. Haragus, G. Iooss: [Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems](http://www.springer.com/us/book/9780857291110), Springer, 2011
might be of some interest to you.
| 3 | https://mathoverflow.net/users/12898 | 228722 | 106,624 |
https://mathoverflow.net/questions/228726 | 12 | Is there any analogue of the Tate curve for (principally polarized) abelian varieties of dimension $g$ ?
| https://mathoverflow.net/users/nan | Analogue of Tate curve for $g>1$ | Yes, due to Mumford. See
Mumford, D.: An analytic construction of degenerating abelian varieties over complete rings, *Comp.. Math.*
**24**, 239-272 (1972).
Of course, there's been plenty of work done since then. See for example the survey:
W. Lutkebohmert,
From Tate's Elliptic Curve to Abeloid Varieties, *Pure a... | 21 | https://mathoverflow.net/users/11926 | 228729 | 106,626 |
https://mathoverflow.net/questions/228731 | 2 | Consider the Fibonacci words $B\_n$:
* $B\_1 = 1$
* $B\_2 = 10$
* $B\_3 = 101$
* $B\_4 = 10110$
* $B\_5 = 10110101$
(start with $B\_1=1$, and go from $B\_n$ to $B\_{n+1}$ by replacing every occurence of $1$ in $B\_n$ with $10$ and every occurence of $0$ with $1$).
The word $B\_n$ has $f\_n$ digits, the $n$-th Fib... | https://mathoverflow.net/users/21339 | Estimate of the number of rabbit integers with a given congruence | You are asking about subsequences of the infinite Fibonacci word [A005614](https://oeis.org/A005614).
One way to describe this sequence is the indices where $\lfloor (n+1) \phi \rfloor \gt \lfloor n \phi \rfloor$, or equivalently when $n \phi \mod 1$ is in a particular interval of length $1/\phi$, namely $[1-1/\phi,... | 7 | https://mathoverflow.net/users/2954 | 228733 | 106,627 |
https://mathoverflow.net/questions/228599 | 6 | For $A,B \subseteq \mathbb{N}$, define $A\sim B$ when there exist *partial* computable functions $f,g\colon \mathbb{N}\rightharpoonup \mathbb{N}$ such that $f$ is defined at least on all of $A$ and $g$ at least on all of $B$ and such that they restrict to inverse bijections $f|\_A\colon A\to B$ and $g|\_B\colon B\to A$... | https://mathoverflow.net/users/17064 | "Partial-computably isomorphic" sets | The equivalence relation $\sim$ is referred to as *recursive equivalence*, and the equivalence classes as *recursive equivalence types*.
I don't know much about them, other than McCarty showed in his PhD thesis *Realizability and Recursive Mathematics*, that in the realizability model $V(\mathcal{Kl})$ this notion co... | 4 | https://mathoverflow.net/users/30790 | 228735 | 106,629 |
https://mathoverflow.net/questions/219216 | 1 | This seems plausible, given the properties of the unit ball of $c\_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be nice to deduce from this that $X$ contains no copy of $c\_0$. I have been searching, but could find no proof either way, a... | https://mathoverflow.net/users/73784 | does every compact convex set in c0 have but countably many extreme points | As stated your question admits an immediate answer because the extreme point structure of finite dimensional convex sets in infinite-dimensional Banach spaces is not related to the structure of the Banach space: for any such set we can find an affine (and thus, preserving extreme structure) map into any other infinite-... | 4 | https://mathoverflow.net/users/85406 | 228741 | 106,630 |
https://mathoverflow.net/questions/228742 | 0 | Consider the iterated function $f^t(z)=f(f(f(...f(z))...))$ where $t \in \mathbb Z$ and $f(z)$ is convergent. Then the iterates of $f(z)$ such as $f^2(z), f^3(z), f^4(z)$ are convergent. Now let $r \in \mathbb Q$ and $s \in \mathbb N$ where $r \times s = t$ and $h(z) = f^r(z)$. Then $h^s(z)=f^t(z)$. Does the convergenc... | https://mathoverflow.net/users/nan | Given $g(h(z))$ is convergent, what can be said about the convergence of $g(z)$ and $h(z)$? | If I understand correctly the setting (say $f$ is a germ at $0\in\mathbb C$ of a biholomorphic function), the answer is no in general, but yes generically. The diffeomorphismn $f$ can always be written as the time-$1$ flow $\exp X$ of a **formal** vector field $X$. The set $$T:=\{t\in\mathbb C : \exp (tX)~ \mathrm{conv... | 5 | https://mathoverflow.net/users/24309 | 228743 | 106,631 |
https://mathoverflow.net/questions/228747 | 2 | I am looking for an example of topological spaces $\langle X\_1,\mathscr{O}\_1\rangle$ and $\langle X\_2,\mathscr{O}\_2\rangle$ such that
there is a homomorphism $h\colon\mathrm{r}\mathscr{O}\_1\longrightarrow\mathrm{r}\mathscr{O}\_2$ between the algebras of regular open sets of respective spaces which fails to preserv... | https://mathoverflow.net/users/22019 | Regular open Boolean algebras and homomorphism which does not preserve nearness of sets | $\textbf{A simple counterexample}$
The mapping $\phi:Ro(\omega+1)\rightarrow Ro(\omega)$ defined by $\phi(U)=U\cap\omega$ is a Boolean algebra isomorphism, but if $E$ denotes the even positive integers and $O$ denotes the odd positive integers, then
$Cl\_{\omega+1}(E)\cap Cl\_{\omega+1}(O)=\{\omega\}$ while
$Cl\_{\om... | 2 | https://mathoverflow.net/users/22277 | 228756 | 106,637 |
https://mathoverflow.net/questions/228758 | 3 | **Problem Setup**
Let $f:A\rightarrow B$, be a continuous function, $A\subset\Re^{n}$,$B\subset\Re^{m}$, $m\geq n$ and $A, B$ compact.
The function $f(\cdot)$ can only be evaluated numerically.
**Question**
Does there exist a general procedure for checking, numerically, whether or not vector functions like $f... | https://mathoverflow.net/users/85422 | Injectivity of vector functions: Numerical Verification | This is hopeless without further assumptions, because no numerical procedure with a finite number of function evaluations can ever rule out a lack of injectivity in parts of the domain where the function hasn't been evaluated. In fact, even strengthening the hypothesis to assume that $f$ is smooth and taking $m=n=1$ is... | 3 | https://mathoverflow.net/users/9022 | 228768 | 106,640 |
https://mathoverflow.net/questions/194557 | 1 | In '*Kriegl, Michor - A convenient setting for global infintite-dimensional analysis*', they say that for an element $x$ in a convenient (i.e. Mackey-complete locally convex) space $X$, a bounded derivation $\partial\_x: C^\infty(X) \to \mathbb{R}$ induces a bounded derivation $\partial\_x: C^\infty(U) \to \mathbb{R}$ ... | https://mathoverflow.net/users/58211 | Restriction of derivations on $C^\infty(X)$ | In 28.1 of your reference an operational tangent vector is defined more carefully than what you write: On the ring $C^\infty(E\supseteq \{x\},\mathbb R)$ of germs at $x$. This induces a point derivation at $x$ on $C^\infty(U,\mathbb R)$ for each $c^\infty$=set containing $x$. To prove the other direction you need assum... | 3 | https://mathoverflow.net/users/26935 | 228771 | 106,642 |
https://mathoverflow.net/questions/228769 | 0 | I was curious if there was a reference which answers the question, What is the maximum number of edges in a graph $G$ with $n$ vertices which does not contain a $5$-cycle? $k$-cycle? The analogous question for $k=4$ is well known.
| https://mathoverflow.net/users/85402 | Reference Request: Graph Edge Density | For cycles of odd length, the only extremal graphs for large $n$ are complete bipartite graphs with the sides as equal as possible. For smaller $n$ there can be other extremal graphs. The complete story was worked out fairly recently by [Füredi and Gunderson](http://arxiv.org/abs/1310.6766).
Cycles of even length are... | 7 | https://mathoverflow.net/users/9025 | 228772 | 106,643 |
https://mathoverflow.net/questions/227600 | 7 | I believe that the circulant Hadamard conjecture (that there are no circulant Hadamard matrices of size greater than $4\times4$) is still open.
I also know that examples of $(n/2) \times n$ matrices which are partial Hadamard circulant have been found experimentally for moderate values of $n.$
To clarify, a partial... | https://mathoverflow.net/users/17773 | Are there infinite constructions for partial circulant hadamard matrices? | Let $r\mbox{-}H(k\times n)$ denote a $k\times n$ partial circulant Hadmard matrix in which a row (and hence all) sums to $r$. It's a known result, see Theroem 9 in [1], that $2\mbox{-}H((p+1)\times 2(p+1))$ exists for any prime power $p$. This is because negacyclic $C$-matrices of order $p+1$ exist. In [2], Paley gave ... | 3 | https://mathoverflow.net/users/14454 | 228775 | 106,644 |
https://mathoverflow.net/questions/228785 | 1 | For each $n$ let $N\_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N\_t\}\_t$ is a family of hypersurface that is evolving with some velocity $V$.
Smooth functions on $N$ have the material derivative
$$Du = \tilde u\_t + \nabla \tilde u \cdot V$$
where $V$ is the velocity of the hyper... | https://mathoverflow.net/users/84288 | material derivative and relation to Riemannian metric | Either you were told wrong or your memory is incomplete
1. All geometric notions of derivatives coincide on smooth *functions*. The metric doesn't enter into it.
2. Your summary of what a material derivative is is not quite correct/complete. What you should have is some version of the following:
Let $M$ denote the... | 2 | https://mathoverflow.net/users/3948 | 228790 | 106,650 |
https://mathoverflow.net/questions/218951 | 8 | This is almost certainly routine to an analyst, so forgive me in advance.
Let $\alpha\_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum\_i \hat{f}(\alpha\_i),$$ where by '$\hat{f}$' I mean the Fourier transform of $f$ regarded as a function on $\mathbb{R}$ (via exte... | https://mathoverflow.net/users/12138 | Can exponential sums be small on a whole interval? | I think I can prove a polynomial bound using complex analysis. This really seems like it shouldn't work but it seems to.
The bound is that for all $\alpha\_1, \dots, \alpha\_n \in \mathbb R$
$$ \sup\_{t \in [A,B]}\left| \sum\_{i=1}^n e( \alpha\_i t) \right| \geq n^{1-\frac{\pi}{4 \arctan {\sqrt{\frac{B}{A}}}-\pi}}... | 6 | https://mathoverflow.net/users/18060 | 228792 | 106,651 |
https://mathoverflow.net/questions/228762 | 14 | Let $a\_n\in \mathbf{N}$ be an infinite sequence such that $\forall i\neq j, a\_i\neq a\_j$.
I have the following theorem:
>
> For $0<c<\frac{3}{2}$, there are infinitely many $k$ for which $[a\_k,a\_{k+1}]>ck$, where $[\cdots]$ denotes least common multiple.
>
>
>
**Idea of proof**: By contradiction. Suppos... | https://mathoverflow.net/users/nan | Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$ | In fact there are sequences $\{a\_k\}$ of pairwise distinct positive integers
such that $[a\_k, a\_{k+1}] \ll k^{1+\epsilon}$ for all $\epsilon > 0$.
We first exhibit a sequence with
$[a\_k, a\_{k+1}] \ll k^{3/2} \log^3 k$ for all $k>1$, which already disproves
the conjecture that $[a\_k, a\_{k+1}] \gg k^2$ infinitel... | 16 | https://mathoverflow.net/users/14830 | 228795 | 106,652 |
https://mathoverflow.net/questions/228774 | 1 | It can easy be shown by induction that the determinant of the $(N-1)\times (N-1)$ matrix
$$\begin{pmatrix}
2 & -1 & & \\
-1 & 2 & \ddots & \\
& \ddots & \ddots & -1\\
& & -1 & 2
\end{pmatrix}$$
equals $N$. (Up to a factor) this matrix corresponds to the representing matrix of the discrete Laplacian on an interval ... | https://mathoverflow.net/users/16702 | Determinant of discrete Laplacian | You already have the answer and proof, so let me just use this post to advertise some basic graph theory :). Take a cycle graph on $N$ vertices, and weigh each edge by $\Delta\_i$, $i=1,2,\dots N$. Then your matrix is a cofactor of the [Laplacian matrix](https://en.wikipedia.org/wiki/Laplacian_matrix) of this graph. By... | 3 | https://mathoverflow.net/users/2384 | 228797 | 106,653 |
https://mathoverflow.net/questions/228794 | 5 | For $x \gt 0,$ what is the greatest $y$ such that $$\sum\_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the series is $$\sum\_ {k=1}^{\infty} \frac{H(k)}{k^y},$$ where $H(k)$ is a harmonic number.
| https://mathoverflow.net/users/61426 | When does this interesting sum diverge? |
>
> In short,
> $$ \begin{cases}
> \text{when }1\leq x & \text{series diverges when }y\le1\\
> \text{when }\frac{1}{2}<x<1 & \text{series diverges when }y\leq\frac{x}{2x-1}\\
> \text{when }0<x\leq\frac{1}{2} & \text{series always diverges.}
> \end{cases}
> $$
>
>
>
When $x>1$, the inner sum of $$\sum\_{k=1}^{\... | 11 | https://mathoverflow.net/users/12176 | 228798 | 106,654 |
https://mathoverflow.net/questions/228805 | 6 | In applications one often encounters very large matrices that barely fit in computer memory, if at all. Naturally one wishes to represent those matrices as compactly as possible. Sometimes one even sacrifices accuracy for compactness, as in [L-BFGS](https://en.wikipedia.org/wiki/Limited-memory_BFGS) for example.
On t... | https://mathoverflow.net/users/38448 | Kolmogorov complexity for matrices | [Kolmogorov’s complexity for positive definite matrices](http://www.sciencedirect.com/science/article/pii/S0024379501003548)
>
> Based on Kolmogorov’s idea, complexity of positive definite matrices
> with respect to a unit vector is defined. We show that the range of
> the complexity coincides with the logarithm ... | 4 | https://mathoverflow.net/users/11260 | 228806 | 106,658 |
https://mathoverflow.net/questions/228787 | 8 | Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a Dehn twist $\tau \in G$ which is not in $H$ but such that $\tau^k$ is in $H$, $k$ being minimal for this property ?
Using th... | https://mathoverflow.net/users/85461 | Subgroups of the mapping class group of a surface generated by Dehn twists | *Edited:*
The paper [Dehn twists have roots](http://arxiv.org/abs/0810.5036) proves that Dehn twists have roots, naturally. The limits on that construction can be found in the paper [Roots of Dehn twists](http://arxiv.org/abs/0906.1601); they bound the degree of the root (linearly, as I recall) in terms of the compl... | 3 | https://mathoverflow.net/users/1650 | 228808 | 106,660 |
https://mathoverflow.net/questions/228812 | 8 | 1) Consider the algebra $\mathcal T (M) = \bigoplus \limits \_{p, q \ge 0} \mathcal T ^{p, q} (M)$, where $\mathcal T ^{p, q} (M)$ is the space of tensor fields of type $(p,q)$. I should endow it with some topology, but I do not know what to choose: the topology of pointwise convergence is one option, the compact-open ... | https://mathoverflow.net/users/54780 | Recovering a smooth manifold from its tensor fields | Look at $M=S^7$. This has 28 smooth structures, and their tangent bundles are all trivial - compare [Parallelizability of the Milnor's exotic spheres in dimension 7](https://mathoverflow.net/questions/58131/parallelizability-of-the-milnors-exotic-spheres-in-dimension-7) . Thus their tensor algebras do not carry more i... | 10 | https://mathoverflow.net/users/35687 | 228817 | 106,661 |
https://mathoverflow.net/questions/228804 | 8 | nLab [defines a strict 2-groups](https://ncatlab.org/nlab/show/strict+2-group) in many different but equivalent ways, among them:
* an internal group object in Cat,
* an internal group object in Grpd
Also, it is known that strict 2-groups may be defined from crossed modules (see for example [Baez](http://arxiv.org/... | https://mathoverflow.net/users/2597 | strict 2-groups VS crossed modules | Let us prove the analogous claim for weak 2-groups (this implies, in particular, the strict case). Let $\mathcal{C}$ be a monoidal category with unit $\mathbb{I} \in \mathcal{C}$.
**Claim**: Suppose there exists a functor $Inv: \mathcal{C} \to \mathcal{C}$ and natural isomorphisms $\psi\_X: \mathbb{I} \stackrel{\con... | 6 | https://mathoverflow.net/users/51164 | 228820 | 106,662 |
https://mathoverflow.net/questions/228819 | 1 | For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple arcs and simple closed curves in E(2) which have positive 2-dimensional Lebesgue measure.
Question (1): Do there exis... | https://mathoverflow.net/users/4423 | Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces | Question 1: Yes. Such curves are called Osgood curves. I couldn't find a construction of an Osgood curve in higher dimensions, but some of the constructions of Osgood curves in the plane generalize. For example, Riesz suggested showing that any totally disconnected set is a subset of a simple curve (see the [Denjoy-Rie... | 1 | https://mathoverflow.net/users/2954 | 228823 | 106,664 |
https://mathoverflow.net/questions/228822 | 4 | I have a modular form I am constructing out of sums and products of various dissected divisor-sum series, namely forms of the type $$f\_i = \sum\_{j=0}^\infty \sigma\_1(36j+i) q^{36j+i}.$$
Each of these can be constructed from the standard forms $\sum \sigma\_1(2n+1) q^{2n+1}$ and $\sum \sigma\_1(3n+1) q^{3n+1}$ by v... | https://mathoverflow.net/users/12878 | How different can characters be for a sum of modular forms to still be in Gamma_0? | The bad news: Yes, you are forced to go to $\Gamma\_{1}(N)$, and not $\Gamma\_{0}(N)$. In particular, let's say that $E = \sum \sigma\_{1}(2n+1) q^{2n+1}$ is the usual weight $2$ level $4$ Eisenstein series. Denote by $E\_{\chi}$, the twist of $E$ by the Dirichlet character modulo $\chi$. Then, you get that
$$
\sum \s... | 3 | https://mathoverflow.net/users/48142 | 228827 | 106,666 |
https://mathoverflow.net/questions/228830 | 7 | In order to know more about product over primes ,I would like to know how do I show that :$$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$$ without using properties of Riemann zeta function ?
**Note01** : it is well known that $$\prod\frac{p^2+1}{p^2-1}=\frac{{\zeta}^2(2)}{\zeta(4)}=\frac{5}{2}$$ but is there other method to ... | https://mathoverflow.net/users/74330 | How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function? | This is a well-known problem, attributed to Sam Wagstaff in Richard Guy's *Unsolved Problems in Number Theory*. Section B48 "Products taken over primes" includes a paragraph
>
> Wagstaff asked for an elementary proof (e.g., without using properties of the Riemann zeta-function that $$\prod\_p \frac{p^2+1}{p^2-1} = ... | 16 | https://mathoverflow.net/users/14830 | 228832 | 106,669 |
https://mathoverflow.net/questions/228826 | 7 | A ***Perron number*** is an algebraic number which is greater than one in absolute value and is greater than all of its Galois conjugates in absolute value as well. Lind's theorem states that any Perron number is spectral radius of some Perron-frobenius matrix (a matrix $A$ is ***Perron-Frobenius*** if all of its entri... | https://mathoverflow.net/users/56571 | lower bound for Perron-Frobenius degree of a Perron number | If a Perron number $\lambda$ has negative trace, then any Perron-Frobenius matrix must have size strictly greater than the algebraic degree of $\lambda$, for example the largest root of $x^3 + 3x^2-15x-46$.
If $B$ denotes the $d\times d$ companion matrix of the minimal polynomial of $\lambda$ (which of course can ha... | 9 | https://mathoverflow.net/users/8112 | 228836 | 106,671 |
https://mathoverflow.net/questions/228815 | 2 | Consider the infinite cartesian product $\Omega=\{0,1\}^{\mathbb{N}}$
as a measurable space endowed with the $\sigma$-algebra $\mathscr{F}$ generated by the cylinder sets and $\sigma:\Omega\to\Omega$ the left shift map. Denote by $\sigma^n(\mathscr{F})$ the $\sigma$-algebra generated by the family of r.v. $\{X\_i(\omeg... | https://mathoverflow.net/users/2386 | Shift Invariance of Backward Martingales for tail trivial probability measures | As discussed in comments, the main question is not well-posed as it stands, since it depends on a choice of representative (mod $\mu$-a.e. equality) for the conditional expectation.
But as to your subsidiary question "Must $\mu$ be shift-invariant on tail events?", the answer is no.
Let $\nu$ be the probability mea... | 2 | https://mathoverflow.net/users/4832 | 228837 | 106,672 |
https://mathoverflow.net/questions/228788 | 5 | Let $X$ be a proper scheme over a field $k$. Let $T$ be a scheme over $k$. Is it true that morphisms $T \times X \to \mathbb{A}^1$ are in bijection with morphisms $T \to \Gamma (X, \mathcal{O}\_X)$ (where the finite-dimensional vector space $\Gamma (X, \mathcal{O}\_X)$ is interpreted as a scheme as usual)?
Or, perhap... | https://mathoverflow.net/users/2095 | Mapping scheme from a proper variety | You can just use all the universal properties one at a time. We only need that $X \to \operatorname{Spec} k$ is qcqs and that $\Gamma(X,\mathcal O\_X)$ is finite-dimensional; both these assumptions are satisfied if $X$ is proper over $k$. We also need that $T \to \operatorname{Spec} k$ is flat, which is always true if ... | 6 | https://mathoverflow.net/users/82179 | 228841 | 106,673 |
https://mathoverflow.net/questions/228571 | 2 | I tried to prove following statement and use some techniques but I couldn't get result :
Question: If [Non wandering Set](https://en.wikipedia.org/wiki/Wandering_set) is whole space then [Recurent set](https://en.wikipedia.org/wiki/Recurrent_point) is dense??
when $T:X \to X$ is hemeomorphism on compact metric spa... | https://mathoverflow.net/users/31058 | If the non wandering set is the whole space, then the recurrent set is dense? | In Khanickus's answer, it is not clear if $T^{n\_i}v\_i=v$ with $v\_i\to v$ can guarantee $T^{n\_i}v\to v$, since $n\_i\to\infty$, and the family $\{T^n:n\ge 1\}$ may not be uniformly continuous. We need a small modification:
1. Given an open set $V\_1\subset X$, pick $n\_1\ge 1$ such that $T^{-n\_1}V\_1\cap V\_1\neq... | 3 | https://mathoverflow.net/users/11028 | 228845 | 106,675 |
https://mathoverflow.net/questions/228799 | 3 | Good day to everyone! Does anybody know if there are *upper bound* estimates for Willmore energy for a given surface?
| https://mathoverflow.net/users/85466 | Upper bound for Willmore energy | Of course, there are many surfaces for which the Willmore energy can be computed explicitely, for example the Clifford torus and for all Willmore spheres.
An important class of surfaces where one gets upper bounds for the Willmore energy are the Lawson surfaces $\xi\_{k,l}$: for $k\geq l$ one has
$$W( \xi\_{k,l})< 4\... | 4 | https://mathoverflow.net/users/4572 | 228855 | 106,678 |
https://mathoverflow.net/questions/228853 | 1 | Let $K$ be a field and $D$ be a central division algebra over $K$ of degree $n$. Suppose that $L\subset D$ is a maximal subfield, so that $[L:K]=n$. Then we know that $L$ is a splitting field, so there exists an isomorphism $f:D\otimes\_KL\to M\_n(L)$ of $L$-algebras.
My question is: does it always exists an $f$ as a... | https://mathoverflow.net/users/57609 | Central division algebras and splitting fields | The answer is "yes" if the field extension $L/K$ is Galois : Anton's proof works in that case.
Classical examples where there exists a Galois degree $n$ extension $L/K$ embedded in $D$ are those of global fields (e.g. numbers fields) and local fields (completions of global fields). In this case the division algebras ... | 3 | https://mathoverflow.net/users/4767 | 228863 | 106,680 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.