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https://mathoverflow.net/questions/188118 | 2 | Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to).
I have a function $F(t)$ which takes a value in $[a, b]$ and returns a random variable on $[0, 1]$ such that $E(F(t)) = P(X <= t)$. I c... | https://mathoverflow.net/users/4959 | Estimating mean and variance of a distribution based on error-prone estimates of its cdf | It turns out that the second approach is actually pretty good, because you can apply Hoeffding's inequality.
Let $R$ be uniform on $[a, b]$. Then $E(1 - F(R)) = \frac{E(X) - a}{b - a}$. So if we sample $T\_1, \ldots, T\_n$ from $a + (b - a)(1 - F(R))$ we have that $P(|\frac{1}{n} \sum T\_i - E(X)| \geq \epsilon (b - ... | 0 | https://mathoverflow.net/users/4959 | 188128 | 93,222 |
https://mathoverflow.net/questions/188061 | 13 | Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ where $F$ vanishes. As is well known, there are $27$ lines on $X$ (in characteristic $\neq 3$).
I had my students veri... | https://mathoverflow.net/users/297 | A funny factorization of the Jacobian coming from the lines on the Fermat cubic | The Jacobian matrix consists of coefficients of $t^3,t^2u,tu^2,u^3$ in the following $4$ partial derivatives $$\partial\_p F(t,u,pt+ru,qt+su) = \partial\_{y}F(t,u,pt+ru,qt+su)t,\\
\ldots\\
\partial\_s F(t,u,pt+ru,qt+su)= \partial\_{z}F(t,u,pt+ru,qt+su)u.$$
Following B. Wellington's comment, (up to permutation of col... | 7 | https://mathoverflow.net/users/14037 | 188135 | 93,223 |
https://mathoverflow.net/questions/187956 | 1 | Let $A$ be a frobenius complement for a group $G$ i.e. $A$ act on $G$ by automorphism s.t. $C\_A(g)=e$ for all nonidentity $g$.
Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. As a result we can see $F[G]$ as a $F[A]$ module. Let $\chi$ be corresponding character of $A$ then
$$\ \chi(g) = \b... | https://mathoverflow.net/users/47344 | Can we say that $A$ is a complement for a group $G$? | As is so often the case, the answer is "not in general". Take $n=1+pq,$ where $p,q$ are primes with $q|p−1$. Let $A$ be a non-Abelian group of order $pq.$ Then $A$ is not a Frobenius complement, but $A$ does have a (complex) character which takes value $n$ on the identity and $1$ everywhere else. The character in quest... | 3 | https://mathoverflow.net/users/14450 | 188139 | 93,225 |
https://mathoverflow.net/questions/188101 | 4 | It is known that for a finite non-abelian simple group $G$ we have $|G|<|C\_G(x)|^3$ for some involution $x$. Is there a better bound for the order of centralizer of a nontrivial element of $G$ (not necessarily involution)? For instance, is there always a nontrivial element $x\in G$ such that $|G|<|C\_G(x)|^2$?
| https://mathoverflow.net/users/40723 | On the size of centralizers in a non-abelian finite simple group | (Later edit: Note that in general, the maximal order of the centralizer of a non-identity element in $G\_{n} = {\rm SL}(2,2^{n})$ is $2^{n}+1,$ while the group order is $(2^{n}-1)2^{n}(2^{n}+1).$ Hence as $n \to \infty$, the limit of $\frac{|G\_{n}|^{\frac{1}{3}}}{{\rm max}\_{x \neq 1}(|C\_{G\_{n}}(x)|)}$ is $1 ).$
I... | 6 | https://mathoverflow.net/users/14450 | 188144 | 93,226 |
https://mathoverflow.net/questions/188149 | 2 | In Andrew, Majda- Vorticity and incompressible flow page 93, there is a theorem which is not proved:
Take a smooth incompressible (free divergence) vector field $v$ in $\mathbb{R}^2$.
Call $w$ its vorticity (its curl), and suppose it is $L^1(\mathbb{R}^2)$.
Then $v$ admits a radial energy decomposition, which means:
... | https://mathoverflow.net/users/48151 | A decomposition of incompressible vector fields | Take $v$ a smooth incompressible vector field and call $w$ its vorticity.
If $w\in L^1(\mathbb{R}^2)$, the number
\begin{align\*}
\alpha := \int\_{\mathbb{R}^2} w \,dx,
\end{align\*}
is well-defined. Looking at Example 2.1. on p.47 (so-called "radial eddies"), we now that for any function $\overline{w}:\mathbb{R}\_+\... | 1 | https://mathoverflow.net/users/27767 | 188151 | 93,227 |
https://mathoverflow.net/questions/188150 | 7 | I'm reading the paper *Loop groups and twisted K-theory I* by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence.
I'm a bit confused with some of their computations, for instance $S^3$ (their example 1.4 in the first section). They take subsets $U\_... | https://mathoverflow.net/users/61887 | Mayer-Vietoris sequence for topological K-theory | Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles on $X$; in particular this holds if $X$ is compact Hausdorff, or if it is a finite-dimensional CW complex. This definition is visibly hom... | 17 | https://mathoverflow.net/users/10366 | 188153 | 93,228 |
https://mathoverflow.net/questions/188138 | 9 | Let $BB(n)$ denote busy beaver function. It's well known that $BB(n)$ dominates all computable functions (I'm quite certain it includes partial computable functions too). However, I was wondering if we can show similar domination over functions computable from some fixed oracle $A$. In particular, I have been wondering... | https://mathoverflow.net/users/30186 | Busy beaver function vs low Turing degrees | It turns out you're asking about the Array Nonrecursive degrees. A degree $\mathbf{a}$ is called Array Nonrecursive (ANR) if for every function $f$ wtt-below $\emptyset'$, there is an $\mathbf{a}$-computable function $g$ which is not dominated by $f$. You don't need to consider every $f \leq\_\text{wtt} \emptyset'$, th... | 9 | https://mathoverflow.net/users/32178 | 188156 | 93,229 |
https://mathoverflow.net/questions/188042 | 0 | A certain functional $T$ is defined as:
$$T(F)=\int\_{(0,1)}F^{-1}(s)M(ds)$$
where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$.
The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of ... | https://mathoverflow.net/users/30510 | Continuity of a Functional | If all you assume is that none of the $M$ assigns mass to a point where $F^{-1}$ is discontinuous then the statement is certainly false. Assume you have a discontinuity at ${1/2}$, it suffices to consider the case where $\mathfrak{M}$ consists of a sequence of Dirac measures located at points that accumulate at $1/2$ f... | 1 | https://mathoverflow.net/users/38566 | 188158 | 93,230 |
https://mathoverflow.net/questions/188092 | 1 | Is there any result about maximal abelian p-subgroups of the exceptional group E\_6(q), where q=p^a is prime power?
| https://mathoverflow.net/users/30252 | abelian p- subgroups of E_6(q) | Aspects of this question have been thoroughly treated in *The Classification of Finite Simple Groups*, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalley groups. In your notation (which differs somewhat from theirs), the $p$-rank is $16a$. This gives only the rank of a ... | 2 | https://mathoverflow.net/users/4231 | 188176 | 93,236 |
https://mathoverflow.net/questions/186628 | 3 | Let $G/K$ be a symmetric space of a non-compact type, i.e. $G$ is a semi-simple connected Lie group, and $K$ is its maximal compact subgroup. Helgason in his book "Differential geometry and symmetric spaces" defines the Weyl group $W(G,K)$ as follows.
Let $\mathfrak{g}=Lie(G), \mathfrak{k}=Lie(K)$. Let
$$\mathfrak{g... | https://mathoverflow.net/users/16183 | Weyl group of a symmetric space | The answer is in the book you quote (1978 version) on page 325.
| 14 | https://mathoverflow.net/users/62288 | 188177 | 93,237 |
https://mathoverflow.net/questions/188160 | 3 | Has anyone seen any function of the following type?
$$
g(x):=\sum\_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.
$$
The question is whether for some constant $c>0$,
$$
\lim\_{x\rightarrow\infty}\frac{1}{x}\log g(x) \ge c.
$$
Thanks a lot for any hints!
| https://mathoverflow.net/users/36814 | Does this function have any exponential growth? | This function doesn't have exponential increase. Take $N=N(x)\sim \ln x$, with a big implied constant. Then $C^n\exp(-a^n/x)$ is small for $n\ge N$, so this part of the series cannot have the same size as an exponential function.
However, for $n\le N$, the terms of the exponential series are still increasing in $n$, ... | 4 | https://mathoverflow.net/users/48839 | 188184 | 93,240 |
https://mathoverflow.net/questions/188193 | 9 | **Motivation:** In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the *quotient* category $\mathcal{K}$ of $\mathcal{A}$-chain complexes modulo homotopy and observes this is triangulated; finally, the derived categor... | https://mathoverflow.net/users/18238 | Localizations or quotients of categories? | They are completely different constructions which are in general absolutely unrelated but might coincide on some rare occasion. the image of closed immersion vs open immersion is quite good I think, especially if you think about it in terme of localization of a ring: it might happen that localizing a ring is the same a... | 9 | https://mathoverflow.net/users/22131 | 188199 | 93,245 |
https://mathoverflow.net/questions/188196 | 1 | Given $n\geq 2$, let us consider the n-cube $H\_n=(V,E)$, i.e. vertex set $V$ is $\{0,1\}^n$.
Here, the edges of $H\_n$ are directed, oriented by set inclusion, i.e., $(x,y)\in E$ iff $x\subseteq y$ and $\text{dist}\_{Hamm}(x,y)=1$).
Let $0^n := (0, 0, \ldots, 0)$ and $1^n := (1, 1, \ldots, 1)$.
Let $S\subseteq V\setmi... | https://mathoverflow.net/users/16758 | n-cube connectivity problem | Every vertex removed from the $k$th layer prohibits $k!(n-k)!\leq (n-1)!$ paths. Thus, if $n$ removed verices prohibit all $n!$ paths, then each of them prohibits exactly $(n-1)!$ paths, and the sets of prohibited paths are disjoint. This means that either all the vertices are on the first layer, or all the vertices ar... | 3 | https://mathoverflow.net/users/17581 | 188200 | 93,246 |
https://mathoverflow.net/questions/188131 | 2 | I am reading Ruiz's *Mapping Degree Theory*, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies
* Normality
* Additivity
* Homotopy invariance
And I have learned the degree theory of differential manifold, which also has the ... | https://mathoverflow.net/users/36119 | Axiomatization of Degree Theory | I have thought about the same question when I am writing my lecture notes on degree theory. Here, I provide some information on this problem and I wish it may somehow help you.
I found that in the book "Differential Topology and General Equilibrium with Complete and Incomplete Markets", you can read from google book... | 3 | https://mathoverflow.net/users/26608 | 188204 | 93,247 |
https://mathoverflow.net/questions/188203 | -2 | Is it known whether there is any example of a pair of rotations in $SO(3)$ about orthogonal axes such that the group that they generate is not a free product of the two cyclic groups generated by each generator?
| https://mathoverflow.net/users/15482 | finitely generated subgroups of SO(3) | Yes, there are many infinite examples in a paper of Radin and Sadun, some of which were rediscovered in a paper of mine via different methods. For example, the group
$$\langle \left( \begin{array}{clcr} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 &0& 1 \end{array} \right),
\left( \begin{array}{clcr} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{... | 2 | https://mathoverflow.net/users/14450 | 188205 | 93,248 |
https://mathoverflow.net/questions/188190 | 6 | This question relates to [Realizing groups as automorphism groups of graphs.](https://mathoverflow.net/questions/37356/realizing-groups-as-automorphism-groups-of-graphs)
Given a [monoid](http://en.wikipedia.org/wiki/Monoid) $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to... | https://mathoverflow.net/users/8628 | Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$ | Hedrlín and Pultr proved that every monoid was the endomorphism monoid of a graph. See their paper "Symmetric relations (undirected graphs) with given semigroup" Monatsh. Math 69 (1965), [eudml](https://eudml.org/doc/177296), DOI: [10.1007/BF01297617](https://doi.org/10.1007/BF01297617).
| 10 | https://mathoverflow.net/users/1266 | 188211 | 93,249 |
https://mathoverflow.net/questions/188194 | 4 | The following paper provides result on an analog of twin primes conjecture for $\Bbb F\_q[X]$
<http://pollack.uga.edu/twins.pdf>
Is there an analog of twin primes conjecture for $\Bbb Z[X]$?
An analog similar to that in the paper of the form "Given $g(X)\in\Bbb Z[X]$, $\forall i\in \Bbb N,\mbox{ }\exists f\_i(X)\in... | https://mathoverflow.net/users/10035 | Twin primes for polynomials in $\Bbb Z[X]$ | Use the Chinese remainder theorem to construct infinitely many $f\_i$ such that $f\_i$ is Eisenstein at one prime, and $f\_i+g$ is Eisenstein at another.
| 10 | https://mathoverflow.net/users/35416 | 188213 | 93,250 |
https://mathoverflow.net/questions/188191 | 1 | Let $G$ be a group and let $\Gamma\_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}\_G(k)=\Gamma\_G(k)/\Gamma\_G(k+1)$ and $$\mathcal{L}\_G:=\bigoplus\_{k\geq 1}\mathcal{L}\_G(k).$$ Then $\mathcal{L}\_G$ has a graded Lie algebra structure induced from the commutator brac... | https://mathoverflow.net/users/15770 | Does the associated Lie algebra determine a group? | It seems to me that even $Q\_8$ and $D\_8$ have isomorphic algebras. In fact, one can construct lots of similar pairs of $2$-nilpotent $2$-groups: roughly, the Carnot algebra captures a symplectic form only (which are all isomorphic), whereas the group itself depends on a quadratic extension of this form, and those dif... | 3 | https://mathoverflow.net/users/44953 | 188215 | 93,251 |
https://mathoverflow.net/questions/188207 | 10 | Let $U,V$ be two nonempty connected open sets in $\mathbb{R}^2$ and $U\subsetneqq V$.I want to ask if there must exist an open ball $B\subset V$ such that $B\not\subset U$ and $B\cap U$ is a nonempty connected open set.
| https://mathoverflow.net/users/58096 | a question about connected open sets in $R^2$ | Let $V$ be the complement of a point $a$, and $U$ the complement of a ray $r$ with end-point $a$. Every ball not containing $a$ and intersecting $r$ is in fact split by $r$.
| 13 | https://mathoverflow.net/users/12705 | 188219 | 93,254 |
https://mathoverflow.net/questions/188217 | 1 | I am trying to understand the proof of lemma 3.1, [in this paper](http://arxiv.org/pdf/0903.2413v1)
In proof, they say that $g(dz\_i,d\tau\_k)=dz\_i(\nabla\tau\_k)=0$ I don't understand first and second equality.In second they say, $g(dz\_i,\theta\_k)=0$ by using $J(dz\_i)={\sqrt {-1}}dz\_i$, but how?
Also, why $\n... | https://mathoverflow.net/users/56353 | Moment map coordinates in tours action | In the last paragraph of the previous page, they say that $\nabla \tau\_k$ is the gradient of $\tau\_k$. This means that if $\tilde g : TM \to T^\*M$ denotes the isomorphism induced by the metric (i.e. $\tilde g(v)(w) = g(v,w)$) then
$$
\tilde g (\nabla \tau\_k) = d\tau\_k.
$$
Then
$$
g(dz\_i, d\tau\_k) = g(\tilde g^{-... | 2 | https://mathoverflow.net/users/4622 | 188222 | 93,256 |
https://mathoverflow.net/questions/188178 | 4 | Let $H$ and $K$ be two proper non-trivial subgroups of the
alternating group $A\_n; n\geq 5$.
Then there exists a maximal subgroup $M$ of $A\_n$
such that $H\not\leq M$ and $K\not\leq M$.
To see this
let $\Omega:=\{1,2,\cdots,n\}$
and as always ${\rm Supp}\_{\Omega}(H):=
\{\omega\in\Omega\mid \omega^{h}\neq \omega,... | https://mathoverflow.net/users/27962 | Generalization of a property of $A_n; n\geq 5$ | Yes, such an extensions to other simple groups exists: Let the finite group $S$ act primitively of degree $n$. We may assume that $H$ and $M$ are cyclic, generated by $x$ and $y$, respectively. We are done unless $x$ and $y$ have disjoint supports. But then one of these elements moves at most $n/2$ elements.
[Guralni... | 7 | https://mathoverflow.net/users/18739 | 188229 | 93,260 |
https://mathoverflow.net/questions/188214 | 3 | $\Sigma\_{3}KP\omega$ be Kripke-Platek set theory with infinity and $\Sigma\_{3}$-separation and $\Sigma\_3$-collection. What strengthening of Barwise's *Definition by $\Sigma$ Recursion* (Theorem 6.4 on page 26 of Admissible Sets and Structures) is supported by $\Sigma\_{3}KP\omega$?
| https://mathoverflow.net/users/37385 | A question on recursion in Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection | $\newcommand{\dom}{\operatorname{dom}\nolimits}\newcommand{\res}{\mathop{\upharpoonright}}$
The general $\Sigma\_n$ case is not simpler than the $\Sigma\_3$ case.
Suppose $G(\bar{x},y,z)$ is a $\Sigma\_n$-function. We want a $\Sigma\_n$-function $F(\bar{x},y)$ such that
$$F(\bar{x},y) = G(\bar{x},y,\{\langle u,F(\bar... | 4 | https://mathoverflow.net/users/2000 | 188230 | 93,261 |
https://mathoverflow.net/questions/188224 | 6 | Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site.
One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start with the projective model structure on simplicial presheaves and then we take the Left Bousefield localization with respe... | https://mathoverflow.net/users/22131 | Site dependance of the Cech weak equivalences on simplicial sheaves | It is independent, provided the two sites of definition, also yield the same infinity topos (so e.g. when they both have finite limits, or the infinity-topos is hypercomplete). Given two simplicial objects $F$ and $G$ of the topos of $\mathcal{E}$, so $$F,G:\Delta^{op} \to \mathcal{E},$$ consider the embedding $$\theta... | 4 | https://mathoverflow.net/users/4528 | 188233 | 93,264 |
https://mathoverflow.net/questions/188239 | 4 | Is it true that every projective sub-variety of degree $d$ in $\mathbb CP^n$ is an intersection of some number of hypersurfaces of degree $d$? Is there some simple proof of this fact? (I believe this is so)
| https://mathoverflow.net/users/13441 | Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$ | It is true. The idea is as follows : suppose your variety, say $X$, has dimension $m$. For any $(n-m-2)$-plane $\pi \subset\mathbb{P}^n$, take the join $\langle \pi ,X\rangle$, that is, the union of the lines $\langle p ,x\rangle$ for $p\in\pi $, $x\in X$. It is easy to see that it is a hypersurface of degree $d$ when ... | 9 | https://mathoverflow.net/users/40297 | 188243 | 93,266 |
https://mathoverflow.net/questions/26212 | 34 | A mango made me wonder about this. (See also [this question](https://mathoverflow.net/questions/24054/), which is in a similar spirit.)
Fix $L >0$ and a smooth body (possibly nonconvex—pears or bananas are fair game!) $B \subset \mathbb{R}^3$ (and assume w/l/o/g below that $L$ is sufficiently large since we can dilat... | https://mathoverflow.net/users/1847 | What is the best way to peel fruit? | Sorry, I'm still not allowed to comment. So I use the "Answer" window...
I'm not completely sure to understand your formulation, but for the case of a 2-dimensional sphere and some fixed width of the pealing, you may find your answer in the sphere-filling ropes of Gelrach and von der Mosel. These are ropes with a cer... | 11 | https://mathoverflow.net/users/61106 | 188247 | 93,268 |
https://mathoverflow.net/questions/188246 | 10 | EDIT: Thanks for your answers and comments. There is indeed a classical easy formula, given by Pietro Majer (with a simple nice proof) in his answer below.
---
Given $x\in\mathbb{R}^n$, $x\_i$ denotes its $i$-th coordinate. My question is:
* What is $Vol(\{x\in[0,1]^n|\sum\_{i=1}^n x\_i\le t\})$ for $t\in\mathb... | https://mathoverflow.net/users/29491 | What's the volume of $\{x\in[0,1]^n|\sum x_i\le t\}$ for real $t$? | The volume $V\_n(t)$ is the cdf of the sum of $n$ independent random variables, uniformly distributed on $[0,1]$. So the density $V\_n'(t)$ is the $n$-fold convolution of the characteristic function of the unit interval, $\chi\_{[0,1]}$. (In particular $V\_n'\in C^{n-2,1}(\mathbb{R})$, with support in $[0,n]$, and poly... | 9 | https://mathoverflow.net/users/6101 | 188256 | 93,272 |
https://mathoverflow.net/questions/188145 | 5 | In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, which are not valuation domains? (Of course any local 1-dimensional domain is such, so non-noetherian almost perfect domain... | https://mathoverflow.net/users/25602 | Integral domains with totally ordered spectra | A good source of such rings consist of the so-called "pseudo-valuation domains" of Hedstrom and Houston. A link to their original (1978) article is: <http://projecteuclid.org/download/pdf_1/euclid.pjm/1102810151>. Among other things in that article, they show the following facts that are relevant to your situation:
1... | 5 | https://mathoverflow.net/users/19045 | 188260 | 93,273 |
https://mathoverflow.net/questions/188258 | 1 | We know from Morse theory that smooth manifold(with or without boundary) is a handlebody.
However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, N. G.(2-AOS2); Burago, Yu. D.(2-AOS2); Zalgaller, V. A.(2-AOS2). They proved
>
> Every 3-dimensional compact Rieman... | https://mathoverflow.net/users/47336 | Handle body of 3-manifold with boundary | 3-dimensional handlebodies are obtained by gluing 1-handles to a 3-ball. It is true, that every compact, orientable 3-manifold has a handle decomposition, but this needs not just 1-handles but also 2- and (in the closed case) 3-handles.
In fact, what you get from a handle decomposition (with w.l.o.g. just one 0- and... | 3 | https://mathoverflow.net/users/39082 | 188263 | 93,274 |
https://mathoverflow.net/questions/188255 | 3 | For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum
$$ S\_p(a,m) = \sum\_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(a x + x^{-1})\right), $$
where $x^{-1}$ is the multiplicative inverse of $x$ modulo $p$. I know $\Im S\_p(a,p)=0$ and that there is the Weil bound
$$ |\Re S\_p(a,p)| \leq 2\... | https://mathoverflow.net/users/10423 | Bounds on imaginary parts of partial Kloosterman sums? | The partial sums, normalized by $\sqrt{p}$, are unbounded, as one varies $a$ over all invertible classes modulo $p$n and lets $p$ go to infinity. This follows from results in [this paper](https://arxiv.org/abs/1410.7892), which also has more precise information on the distribution of the partial sums, including real an... | 6 | https://mathoverflow.net/users/57842 | 188268 | 93,277 |
https://mathoverflow.net/questions/188269 | 1 | Let $(\Omega, \mathcal F\_1, {\mathbb P}, \mathbb F = \{\mathcal F\_t\}\_{0\le t \le 1})$ is a
filtered probability space. Let $L^2\_{\mathbb F}$ be a collection of all $\mathbb F$ progressive measurable
1-d processes $x$, such that
$$\mathbb E [\int\_0^1 x^2(s) ds ] <\infty.$$
Define
$$\langle x, y \rangle = \mathbb ... | https://mathoverflow.net/users/5656 | Does $L^2$ progressive measurable processes form a Hilbert space? | The answer is indeed yes, as every adapted measurable real-valued process admits a progressively measurable modification. This is classical Theorem due to Paul-André Meyer (cf. Paul A. Meyer. Probabilities and Potential, Blaisdell 1966; Theorem T46, p. 68)
| 1 | https://mathoverflow.net/users/20026 | 188273 | 93,278 |
https://mathoverflow.net/questions/188289 | 7 | Let $n$ be a positive integer. A subset of $[n] := \{1,2,...,n\}$ having $k$ elements will be called a $k$-subset.
For $n,k \in \mathbb{N}$ with $k \leq \lfloor n/2 \rfloor$, it is clear that one can associate bijectively $(n-k)$-subsets with $k$-subsets through complements: $S \mapsto \bar{S}$. However, the image is... | https://mathoverflow.net/users/62334 | Inclusion-preserving bijection between subsets of cardinality k and n-k | Each $(n-k)$-subset contains $m$ $\,k$-subsets, and each $k$-subset is contained in $m$ $\,(n-k)$-subsets, where $m=\binom{n-k}k$. An $m$-regular bipartite graph can be decomposed into $m$ perfect matchings.
| 9 | https://mathoverflow.net/users/43266 | 188294 | 93,285 |
https://mathoverflow.net/questions/188287 | 2 | To make the question more precise:
We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$.
Let $\mathcal{C}$ be a collection of compact and zero-dimensional topologies on $\mathbb{N}$ such that for $\tau\_1 \neq \tau\_2 \in \mathc... | https://mathoverflow.net/users/8628 | How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$? | By a theorem of Mazurkiewicz and Sierpinski, every countable compact Hausdorff space is homeomorphic to an ordinal, so there are at most $\aleph\_1$ of them. Conversely, any countable ordinal can be the Cantor-Bendixson rank of a countable compact Hausdorff space (easy to prove by induction), so there are indeed $\alep... | 7 | https://mathoverflow.net/users/75 | 188295 | 93,286 |
https://mathoverflow.net/questions/188288 | 11 | A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.
Is there any example of bounded operator acting on a complex non-complete normed space with empty spectrum?
I understand that the spectrum of an operator $T$ is the set of all... | https://mathoverflow.net/users/39421 | Bounded operator on a normed space with empty spectrum | Take an operator on a Banach space whose image is dense, whose spectrum is $\{0\}$ but that has no kernel, for example
$$
T(f)(x)=\int\_x^1f(y)dy
$$
acting on $H:=L^2([0,1])$.
Then its restriction to the dense subspace $D:=\bigcap\_n Im(T^n)$ should have the property you desire.
First of all, $D$ is dense in $H$ ... | 14 | https://mathoverflow.net/users/5690 | 188299 | 93,288 |
https://mathoverflow.net/questions/188236 | 8 | **Question:** Is it consistent (relative to the existence of large cardinals) that there is an elementary embedding $j\colon V\to M$ (where $M$ is transitive model) that factors as $j = j\_n \circ k\_n$ for $n < \omega$, such that for every $n$, $\text{crit }j\_n = \kappa$ (the same ordinal), but $j\_{n} (\kappa) < j\_... | https://mathoverflow.net/users/41953 | Elementary embeddings with the same critical point | Yes, this situation can occur. One should simply undertake the dual of the construction you had suggested with iterated ultrapowers.
Specifically, suppose that $\mu$ is a normal measure on a measurable cardinal
$\kappa$ and let $j:V\to M\_\omega$ be the embedding arising from
iterating the ultrapower $\omega$ many t... | 4 | https://mathoverflow.net/users/1946 | 188303 | 93,289 |
https://mathoverflow.net/questions/188201 | 2 | Let $\tilde{B\_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to intersect itself. Similarly let $\tilde{P\_n}$ be the homotopy pure braid group.
I am reading *A study of Braids* by K. Murasugi and B. I. Kurpita. In Chapter 7, Theorem 3.3, a presentation of $\tilde{B\... | https://mathoverflow.net/users/15770 | Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups? | Consider $A\_{2,3}=\sigma\_2^2$.
We have that $\sigma\_2^2$ does not commute with $\sigma\_1^2.\sigma\_2^2.(\sigma\_1^2)^{-1}$ in the homotopy pure braid group, i.e, $[\sigma\_1^2,\sigma\_2^2]$ is non trivial.
[Habegger and Lin](http://math.ucr.edu/~xl/papers/habegger-lin.pdf) showed that the homotopy pure braid gr... | 3 | https://mathoverflow.net/users/62345 | 188315 | 93,294 |
https://mathoverflow.net/questions/188202 | 6 | An abstract $L\_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L\_1[0,1]$ is a separable example that contains subspaces isomorphic to $L\_p[0,1]$ for $1<p\leq 2$. The space of Borel measures $M[0,1]$ is a non-separable example, but each reflexive subspace of $M[0,1... | https://mathoverflow.net/users/39421 | Reflexive subspaces of non-separable abstract $L_1$ spaces | I guess so. Here's an outline which reduces everything to the separable case.
Let $p\in (1,2]$. Then of course $L\_p[0,1]^I$ is finitely representable in $L\_1$. Take an ultrafilter $U$ (on some ridiculously large index set) such that $L\_p[0,1]^I$ is a subspace of $L\_1^U$. By Kakutani's theorem, $L\_1^U = L\_1(\mu... | 5 | https://mathoverflow.net/users/15129 | 188331 | 93,297 |
https://mathoverflow.net/questions/84915 | 5 | Given a circular unitary ensemble, with the following joint density:
$p(\theta\_1,\ldots, \theta\_n) = Z\_n \prod\_{j < k} |e^{i \theta\_j} - e^{i \theta\_k}|^2$,
is the following statement true? With high probability the eigenvalues are within distance $\mathcal{O}(1)$ from the evenly spaced set of $n$ points $(0... | https://mathoverflow.net/users/4923 | rigidity of eigenvalues of circular ensemble | If I understand correctly, what you are looking for is Lemma 10 (when m=1) in <http://arxiv.org/pdf/1210.2681v3.pdf> by Elisabeth and Mark Meckes.
| 2 | https://mathoverflow.net/users/15517 | 188337 | 93,299 |
https://mathoverflow.net/questions/188335 | 11 | I seem to see three different things that are being called the Laplacian of a graph,
* One is the matrix $L\_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ is the (possibly signed) adjacency matrix.
* The other is to say that the matrix $L\_2$ is a $\vert V \vert \times \... | https://mathoverflow.net/users/36554 | Are these three different notions of a graph Laplacian? | These are usually known as the Laplacian, the normalized Laplacian and the unsigned Laplaian. All three are positive semidefinite. If the graph is regular, they all provide the same information.
If the graph is not regular they are, in general, independent. The normalized Laplacian is the right tool for the analysis ... | 17 | https://mathoverflow.net/users/1266 | 188342 | 93,300 |
https://mathoverflow.net/questions/188339 | 5 | What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible $\Delta^1\_3$ set of integers if one assumes the nonexistence of $0^\sharp$?
| https://mathoverflow.net/users/17218 | Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$? | It is consistent, relative to just $ZFC$, that there exists a non-constructible $\Delta\_3^1$ set of integers.
Such a result was first produced in the paper ``Some allpications of almost disjoint sets'' by Jensen-Solovay, using almost disjoint forcing.
In fact we can prove the following stronger result which is due t... | 8 | https://mathoverflow.net/users/11115 | 188347 | 93,302 |
https://mathoverflow.net/questions/188336 | 5 | Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through the local diffeomorhism $Z:\Gamma \times \mathbb{R} \to M$
$ Z(y,z) = Exp\_y (z N(y)) $ where $N$ is the unit vector al... | https://mathoverflow.net/users/50438 | Minimal surfaces + Semi-Geodesic Coordinates | Yes, this can always be done as long as $\Gamma$ has a positive injectivity radius.
In fact, calculation yields the following formula: Let $V$ be the unit vector field (defined in a neighborhood of $\Gamma$) that is $g$-perpendicular to the level sets of $t$ (i.e., $V$ is perpendicular to each $\Gamma\_t$).
This $V$... | 5 | https://mathoverflow.net/users/13972 | 188360 | 93,306 |
https://mathoverflow.net/questions/188344 | 3 | What is an example of a simple $C^{\*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?
| https://mathoverflow.net/users/36688 | Simple $Z^{*}$ algebra | Take compact operators on a non-separable Hilbert space. For each such an operator you'll find a rank-one projection with range in the kernel of that operator.
~~I guess that by some Löwenheim–Skolem-type argument, this gives you a separable example too.~~ (As observed by Andreas Thom.)
| 6 | https://mathoverflow.net/users/15129 | 188362 | 93,308 |
https://mathoverflow.net/questions/188386 | 1 | Suppose $X$ is a projective variety, and $D$ is a Cartier divisor on $X$. Is it possible (or this is always true) that there is a (ramified) double cover $\pi: \tilde{X} \to X$ such that $\frac{1}{2}\pi^\*(D)$ is also a Cartier divisor?
| https://mathoverflow.net/users/29730 | Cartier divisor on a double cover | This is not always possible. Take $X=\mathbb{P}^2$, $D=$ a line. If $\pi ^\*D$ is twice a Cartier divisor on $\tilde{X} $, this holds also on the normalization of $\tilde{X} $, so you can assume that $\tilde{X} $ is normal. Then $(\frac{1}{2}\pi ^\*D)^2 = \frac{1}{2}D^2=\frac{1}{2}$; but on a normal surface the interse... | 8 | https://mathoverflow.net/users/40297 | 188389 | 93,316 |
https://mathoverflow.net/questions/188364 | 5 | To which extent the adjunction $F\dashv N\_\omega$ generated by the $\omega$-nerve described at [$n$Lab - oriental](http://ncatlab.org/nlab/show/oriental) (obtained as a particular instance of the nerve-realization paradigm) is linked to the adjunction generated by the functor $O\_{[\Theta]}\colon \Theta\to \textbf{Str... | https://mathoverflow.net/users/7952 | $\omega$-nerve versus $\Theta$-nerve | As you remark, the cellular and simplicial nerves arise from the functors $J \colon \Theta \longrightarrow \omega\text{-Cat}$ and $O \colon \Delta \longrightarrow \omega\text{-Cat}$ respectively, where $J$ is the full inclusion of Joyal's cell category and $O$ is Street's orientals functor.
A significant difference ... | 8 | https://mathoverflow.net/users/57405 | 188410 | 93,320 |
https://mathoverflow.net/questions/188405 | 14 | The $n=3$'rd Catalan number ([A000108](http://oeis.org/A000108)) is $1,1,2,5$ : $\frac{\binom{2n}{n}}{n+1}=\frac{\binom{6}{3}}{4}=\frac{20}{4}=$ ***5***.
The $n=4$'th Fibonacci number ([A000045](http://oeis.org/A000045)) is $1,1,2,3,5,...$ : ***5***.
>
> ***Q***. Which other Fibonacci numbers (besides $\{1,2,5\}$... | https://mathoverflow.net/users/6094 | When does a Catalan number equal a Fibonacci number? | A result of the type you seek follows easily from [Carmichael's theorem](http://en.wikipedia.org/wiki/Carmichael%27s_theorem), that if $m > 12$, then there is a prime $p$ that divides $F\_{m}$, but does not divide $F\_{k}$ for $k < m$.
Suppose $C\_{n} = \binom{2n}{n}/(n+1)$ and we assume that $C\_{n} = F\_{m}$. All ... | 38 | https://mathoverflow.net/users/48142 | 188412 | 93,322 |
https://mathoverflow.net/questions/188403 | 7 | I'm sure this is well-known, but: suppose I have a non-constructible real $r\in V-L$. Under what conditions is there a poset $\mathbb{P}\in L$ and a $G$ which is $\mathbb{P}$-generic over $L$, such that $r\in L[G]$? (Note: $G$ is not necessarily a real.)
Obviously it is consistent (e.g., with $V=L$) that the answer i... | https://mathoverflow.net/users/8133 | When can we reach a real by forcing? | Monroe's observation about $0^\sharp$ is certainly a natural example, but let me point out that in fact one doesn't need $0^\sharp$ to make such a situation happen. The consistency strength of having a real that is not set-generic over $L$ does not actually go beyond ZFC itself.
To see this, start in $L$ and then un... | 6 | https://mathoverflow.net/users/1946 | 188415 | 93,324 |
https://mathoverflow.net/questions/188417 | 18 | One of the standard problems in convex optimization is the calculation of the maximum entropy distribution that satisfies some set of criteria. For example, if $\mathbf{x} \in \mathbb R^n$ is an unknown vector of probabilities, one might solve a problem like $$\mathrm{maximize}\_\mathbf{x\geq0} -\sum\_{i=1}^n x\_i \log... | https://mathoverflow.net/users/62398 | When do people actually use the maximum entropy distribution? | In finance, finding risk neutral probabilities can be done via max-entropy methods.
In short, you observe prices $p\_i$ of a finite number of instruments $\phi\_i$, and you seek a probability measure $P$ such that $p\_i = \int\phi\_i\,\mathrm{d}P$. There is in general no uniqueness of $P$, and finding such a $P$ can ... | 9 | https://mathoverflow.net/users/62405 | 188435 | 93,331 |
https://mathoverflow.net/questions/188366 | 23 | It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space without isolated points is homeomorphic to $\mathbb{Q}$. (A proof can be found [here](http://at.yorku.ca/p/a/c/a/25.pdf) a... | https://mathoverflow.net/users/46290 | Which ordered fields are homeomorphic to their power? | $\let\ob\overline\let\sset\subseteq\let\nsset\nsubseteq\DeclareMathOperator\dom{dom}$ There are many such fields different from $\mathbb R$.
>
> **Proposition:** Let $F$ be a topological field of cardinality $2^\kappa$ with a dense subfield $D$ of cardinality $\kappa$. Then there exists an intermediate field $D\sse... | 13 | https://mathoverflow.net/users/12705 | 188447 | 93,339 |
https://mathoverflow.net/questions/188422 | 6 | I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of
$$ \int\_{B\_{r}(x)} |f(y) - f(z)|^{p} dy \leq c r^{n+p-1} \int\_{B\_{r}(x)} |Df(y)|^{p} |y-z|^{1-n} dy$$
where $f \in C^{1}(B\_... | https://mathoverflow.net/users/62399 | Poincare-like inequality on compact Riemannian manifolds | The same inequality holds on Riemannian manifolds, at least if you want it for small $r$.
Fix a point $x\in M$ and let $r\_0$ be the injectivity radius at it.
If $0<r\leq\frac12r\_0$, then $\exp\_x:U\_r\to B\_r(x)$ is a diffeomorphism and bi-Lipschitz continuous with a Lipschitz constant independent of $r$.
Here $B\_r(... | 5 | https://mathoverflow.net/users/55893 | 188450 | 93,340 |
https://mathoverflow.net/questions/188451 | 4 | Let $K/\mathbb{Q}$ be an imaginary quadratic extension with discriminant $-D$. Then there is an elliptic curve $E$ over $\overline{\mathbb{Q}}$ such that End$(E)^{0}: =$ End$(E) \otimes Q = K$.
Now let $p$ be a prime number. Is there any result (with some restrictions on $-D$ and $p$) which will make the above state... | https://mathoverflow.net/users/62410 | Elliptic curves with maximal order in an imaginary field | There is such an elliptic curve in char $p$ if and only if $p$ splits in $K$. The elliptic curve in char. zero with $K$ as endomorphisms is CM so has potential good reduction everywhere and the reduction modulo a prime above $p$ is an elliptic curve in char $p$ with $K$ contained in the (field of fractions of) the endo... | 8 | https://mathoverflow.net/users/2290 | 188452 | 93,341 |
https://mathoverflow.net/questions/188436 | 3 | In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A\_{2}$ et let $\alpha,\beta$ be its simple roots.
How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the Verma module $M(\alpha+2\beta)$?
As noted in the Wikipedia article on Verma modules, their definition relies on a stac... | https://mathoverflow.net/users/57464 | How to find the multiplicity of weight in a Verma module? | Information about Verma modules is not so scarce. In any case, the multiplicities of weights are (in principle) quite easy to calculate, which led BGG in their 1971 paper to an elegant proof of the Weyl and Kostant formulas. (My 2008 book *Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$* pu... | 7 | https://mathoverflow.net/users/4231 | 188453 | 93,342 |
https://mathoverflow.net/questions/188416 | 7 | In [ABGHR](http://arxiv.org/pdf/1403.4325.pdf) Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is equivalent to the well known $BGL\_1(\mathbb{S})$. Then the Thom spectrum $Mf$ is defin... | https://mathoverflow.net/users/11546 | Thom isomorphism from the ABGHR perspective | Let's try to see what a lift $\tilde f : X\to R\textrm{-triv}$ is. Recall that $R\textrm{-triv}$ is the $\infty$-groupoid of $R$-lines with a specified isomorphism with $R$, so a lift $\tilde f$ corresponds to giving a natural equivalence of $f$ with the constant functor $X\to R\textrm{-line}$ with value $R$. This is t... | 7 | https://mathoverflow.net/users/43054 | 188461 | 93,346 |
https://mathoverflow.net/questions/188455 | 8 | According to Bryant's work on special holonomy groups, $G\_2\subset SO(7)$ may be defined as the group preserving the following 3-form:
$\phi\_0=\mathrm{d}x\_{123}+\mathrm{d}x\_{145}+\mathrm{d}x\_{167}+\mathrm{d}x\_{246}-\mathrm{d}x\_{257}-\mathrm{d}x\_{347}-\mathrm{d}x\_{356}$
where $\mathrm{d}x\_{ijk}=\mathrm{d}x... | https://mathoverflow.net/users/62367 | $Spin(7)$ as stabilizer of a $4$-form | There are at least two sources for this: First, Harvey and Lawson, *Calibrated geometries* (Acta Math. 1982) proves this (i.e., that the stabilizer of $\Omega\_0$ is isomorphic to the nontrivial double cover of $\mathrm{SO}(7)$) using properties of the octonions. Second you can find a proof that doesn't rely on knowled... | 11 | https://mathoverflow.net/users/13972 | 188469 | 93,347 |
https://mathoverflow.net/questions/188463 | 1 | Let $E$ be the elliptic curve
$$y^2 =x^3 - 19\*67 x$$
and $P=[26011/625,2159616/15625]$, I want to compute $\hat{h}(P)$ using formula given in
Fujita, Y., & Terai, N. (2011). Generators for the elliptic curve $ y^ 2= x^ 3-nx$. Journal de théorie des nombres de Bordeaux, 23(2), 403-416.
To compute Archimedean part ... | https://mathoverflow.net/users/41032 | computing height on elliptic curve of the form $y^2=x^3-nx$ | Is there some reason that you want to use this formula? You can get the height directly in PARI with the commands:
gp > EE=ellinit([0,0,0,-19\*67,0]);
gp > PP=[26011/625,2159616/15625];
gp > ellheight(EE,PP)
%1 = 8.6541981
For the local archimedean height, the following formula is in *Advanced Topics in the A... | 4 | https://mathoverflow.net/users/11926 | 188470 | 93,348 |
https://mathoverflow.net/questions/188495 | 1 | We have two probability spaces $(\Omega\_1,\mathcal{F\_1},P\_1)$ and $(\Omega\_2,\mathcal{F\_2},P\_2)$. Is it possible to construct probability space $(\Omega=\Omega\_1\times\Omega\_2,\mathcal{F},P)$ such that:
1. $\forall A\_1\in\mathcal{F\_1}, A\_2\in\mathcal{F\_2}$ set $A\_1\times A\_2\in\mathcal{F}$ and its measu... | https://mathoverflow.net/users/62421 | Extend product sigma-algebra to cross-constant sets | Consider the case $\Omega\_1 = \Omega\_2 = [0,1]$, equipping both factors with Lebesgue measure. Let $S$ be *any* subset of $[0,1]$. Then the set $$(S \times [0,1/2]) \cup (S^c \times (1/2,1])$$ is "cross-constant" in your sense --- each vertical slice is measurable and has measure $1/2$. So all such sets would belong ... | 3 | https://mathoverflow.net/users/23141 | 188502 | 93,361 |
https://mathoverflow.net/questions/188518 | 10 | Consider the ordinary sphere $\mathbb{S}^2\subset \mathbb{R}^3$ and a spherical triangle $T\subset \mathbb{S}^2.$ I'm looking for a formula from which the perimeter $P$ of $T$ is "computable" given the three interior angles $\gamma\_1,\gamma\_2,\gamma\_3$ (and the area $\vert T \vert$) of the triangles, i.e. i'm search... | https://mathoverflow.net/users/21870 | Formula for the Perimeter of a spherical triangle? | First, you don't need to know the area separately, since that is given by the classic formula
$$
|T| = (\gamma\_1+\gamma\_2+\gamma\_3) - \pi.
$$
Second, if $\ell\_i$ is the length of the side opposite $\gamma\_i$, then the standard spherical trig formula called the *polar law of cosines* gives
$$
\ell\_i = \cos^{-1}\le... | 15 | https://mathoverflow.net/users/13972 | 188523 | 93,365 |
https://mathoverflow.net/questions/188519 | 2 | How to prove that there exist two different programs A and B such that A printing code of B and B printing code of A without giving actual examples of such programs?
Update: We could prove via Kleene's recursion theorem that there is program printing it's own code. I tried to apply similar technics but didn't succeed... | https://mathoverflow.net/users/62438 | Prove existence of different programs printing each other code | Any Turing-complete model of computation will have programs with this property. Specifically, let $\varphi\_e$ denote the function computed by program $e$, in whatever such system you favor. Define two computable functions $h\_1$ and $h\_2$ so that
$$\varphi\_{h\_1(x,y)}(z)=\varphi\_x(x,y)\qquad\text{ and }\qquad\var... | 4 | https://mathoverflow.net/users/1946 | 188527 | 93,366 |
https://mathoverflow.net/questions/188528 | 12 | My question is the following:
Given an infinite dimensional Banach space $E$ and a one-dimensional linear subspace $F\subset E$. It is well-known that this one-dimensional linear subspace is closed and always complemented. So, we have the following topological isomorphism:
$$E \cong F \times E/F.$$
Now, what is t... | https://mathoverflow.net/users/58628 | Banach space modulo a one-dimensional subspace =? | Gowers proved in "A solution to Banach’s hyperplane problem" (1994)
An infinite-dimensional Banach space X is constructed which is not isomorphic to X ⊕ R. Equivalently, X is not isomorphic to any of its closed subspaces of codimension one. This gives a negative answer to a question of Banach. In fact, X has the stro... | 14 | https://mathoverflow.net/users/nan | 188534 | 93,368 |
https://mathoverflow.net/questions/188533 | 16 | I'm searching the wheres and whys about the integral notation for co/[ends](http://ncatlab.org/nlab/show/end). Who was the first to adopt it? Can you give me a precise pointer or tell me the whole story about it? Was s/he motivated by the Fubini rule only, or there is another additional reason?
Thanks a lot!
| https://mathoverflow.net/users/7952 | History of integral notation for coends | From Ross Street's *An Australian conspectus of higher categories*:
>
> Kelly developed the theory of enriched categories describing enriched adjunction and introducing the variety of limit he called *end*. I later pointed out that Yoneda had used this concept in the special case of additive categories using an *in... | 19 | https://mathoverflow.net/users/57405 | 188536 | 93,370 |
https://mathoverflow.net/questions/188515 | 4 | Let $G,H$ be simple undirected graphs without loops such that there are graph homomorphisms $f\_1:G\to H$ and $f\_2: H\to G$.
An easy argument shows that we have $\chi(G) = \chi(H)$, even for graphs with infinite chromatic number.
What other conclusions can be made about $G, H$ if we know that there are graph homom... | https://mathoverflow.net/users/8628 | What can be said about graphs if there are homomorphisms in both directions? | Let $\omega (G)$ be the *clique number* of $G$ i.e. the largest $n$ such that $G$ contains $K\_n$ as a subgraph; this is the same as the largest $n$ such that a homomorphism $K\_n \rightarrow G$ exists. By composing $K\_n \rightarrow G \xrightarrow{f\_1} H$ we obtain $\omega (G) \leq \omega (H)$ and similarly in the ot... | 6 | https://mathoverflow.net/users/60535 | 188538 | 93,371 |
https://mathoverflow.net/questions/188485 | 10 | this relates to a question asked at MSE. i was able to find an answer using complex numbers.
here is the question: there are five points on a circle. take any three points, through the centroid of the three points draw a line orthogonal to the line through the remaining two points. take all ten combinations. it turn... | https://mathoverflow.net/users/14688 | ten concurrent lines | Your solution takes two lines, calculates their intersection, and confirms that it is as you claimed. Here is a proof which uses position vectors to show that each line goes through the specified point $$\mathbf{p}=\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}+\mathbf{e}}{3}.$$ I note at the end that this proof wor... | 13 | https://mathoverflow.net/users/8008 | 188552 | 93,377 |
https://mathoverflow.net/questions/188547 | 0 | I somewhat advanced since [6j symbols trouble when irrep multiplicity >1](https://mathoverflow.net/questions/106812/6j-symbols-trouble-when-irrep-multiplicity-1) (because I found the decades old papers where the Racah algebra is dealt with, e.g. "Coupling Coefficients and Tensor Operators for Chains of Groups",
P. H. B... | https://mathoverflow.net/users/11504 | A 6j multiplicity paradox | The fusion ring you consider is known as (even part of) $E\_6-$subfactor. The corresponding
6j symbols (also called associativity matrices) are given in the [paper](http://arxiv.org/pdf/0704.0208.pdf) by Hagge and Hong, see Appendix.
| 4 | https://mathoverflow.net/users/4158 | 188557 | 93,379 |
https://mathoverflow.net/questions/188567 | 0 | The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?
| https://mathoverflow.net/users/14233 | Example of a polycyclic group which is not of polynomial growth? | Semi-direct product of $\mathbb Z$ and $\mathbb Z^2$ determined by the matrix
$\left( \begin{array}{ccc}
2 & 1 \\
1 & 1 \end{array} \right)$.
| 4 | https://mathoverflow.net/users/8588 | 188568 | 93,383 |
https://mathoverflow.net/questions/188474 | 1 | **Edit:** According to the comment of Andre Henriques I revise the question:
What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative algebra, such that for all unital subalgebra $B$ of $A$, $ K\_{0}(B)$ has $\mathbb{Z}$ as a summand? This question is motiva... | https://mathoverflow.net/users/36688 | A question on K- theory of non commutative $C^\star$ algebra | The question only requires regarding K$\_0$ as an abelian group (it has a natural pre-ordering too), which makes it easy to construct simple (in the technical sense) examples.
Let $A$ be a simple infinite dimensional unital AF C\*-algebra, whose K$\_0$ group is free (as an abelian group) [*lots* of examples exist; s... | 5 | https://mathoverflow.net/users/42278 | 188576 | 93,385 |
https://mathoverflow.net/questions/188574 | 8 | I'm pretty sure that if the function $f$ is continuous on $[x\_1,x\_3]$ and twice-differentiable on $(x\_1,x\_3)$, with $x\_1 < x\_2 < x\_3$, then there must exist $x$ in $(x\_1,x\_3)$ for which $f''(x)$ equals $$\dfrac{((x\_2-x\_1)(f(x\_3)-f(x\_1)) - (f(x\_2)-f(x\_1))(x\_3-x\_1))}{ ((x\_2-x\_1)(x\_3-x\_2)(x\_3-x\_1))}... | https://mathoverflow.net/users/3621 | A generalized mean-value theorem | Let $q(x)$ be the quadratic through the $(x\_i, f(x\_i))$. Set $g(x) = f(x) - q(x)$. Then $g(x\_1) = g(x\_2) = g(x\_3) = 0$ and you want to show $g''(x)=0$ for some $x$. Apply Rolle's theorem to get that $g'$ has zeroes $y\_1$ and $y\_2$ in $(x\_1, x\_2)$ and $(x\_2, x\_3)$ respectively, then apply it again to get that... | 15 | https://mathoverflow.net/users/297 | 188578 | 93,387 |
https://mathoverflow.net/questions/188563 | 3 | It is proved in this [paper](http://www.jstor.org/stable/1971417) by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the $\mathcal{O}\_X$-algebra $$\mathcal{R}\_X(D):=\oplus\_{m\geq 0}\mathcal{O}\_{X}(mD)$$ is finitely... | https://mathoverflow.net/users/29730 | Finite generation of certain $\mathcal{O}_X$-algebra | Indeed, this is fine. In fact it is an exercise in Kollár's arXiv notes (of course utilizing the MMP):
<http://arxiv.org/pdf/0809.2579.pdf>
see in particular exercises 90-110.
The upshot is that everything is fine for KLT ambient spaces.
| 3 | https://mathoverflow.net/users/3521 | 188584 | 93,388 |
https://mathoverflow.net/questions/188596 | 5 | A set $X$ is called *cohesive for $(R\_i)\_{i\in \mathbb{N}}$* if it is infinite and for each $i$ we have $X\subseteq^\* R\_i$ or $X\subseteq^\* \overline{R\_i}$. (Where $X\subseteq^\*Y$ means that $X$ is contained in $Y$ up to finitely many exceptions.)
A set $X$ is called *cohesive* if it is cohesive for all comput... | https://mathoverflow.net/users/3365 | Is below every cohesive set a 1-generic? | No -- each high degree contains a cohesive set. And Cooper showed there is a high minimal degree. But no 1-generic has minimal degree.
| 6 | https://mathoverflow.net/users/4600 | 188600 | 93,390 |
https://mathoverflow.net/questions/188548 | 2 | So generally we define a differential relation to be $\mathcal{R} \subset X^{(r)}.$ In the case that $X=M\times N$ is it possible to have $\mathcal{R}=X^{(1)}$? So in this case the formal solutions would be (my guess) just fiberwise linear bundle maps that covers a continuous map and the genuine (holonomic) solutions w... | https://mathoverflow.net/users/62456 | Question about the h-principle | Yes, it is possible to have $\mathcal{R}=X^{(1)}$, i.e. to have a system of partial differential equations that consists of no equations at all. Then as you say the formal solutions are sections of $X^{(1)} \to M$, while the holonomic solutions are sections of $X \to M$. The surjectivity is obvious, since the fibers of... | 5 | https://mathoverflow.net/users/13268 | 188601 | 93,391 |
https://mathoverflow.net/questions/188579 | 12 | I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. There is one obvious point at $(0, 124416)$. I have some experience in finding rational points on elliptic curves. I also ha... | https://mathoverflow.net/users/62471 | rational points of a hyperelliptic curve | By now there is a fairly rich literature on computing the set of rational points on curves of higher genus, see for example my survey paper on ["Rational points on curves"](http://www.mathe2.uni-bayreuth.de/stoll/schrift.html#AG38).
What one can do for your concrete curve is this. First, one can simplify the curve eq... | 18 | https://mathoverflow.net/users/21146 | 188604 | 93,393 |
https://mathoverflow.net/questions/185822 | 7 | The series
$$\sum\_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$
has come up when I'm trying to apply the methodology in this paper (<http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf>) to Poisson regression.
When $\beta = 1,$ it is $e^{e^c}$ and when $\beta = 2,$ it is $I\_0(2e^c),$ the modified Bessel functio... | https://mathoverflow.net/users/2586 | expression for infinite series with powers of factorial in denominator | I copied this from my comments above:
This sum was discussed in the book "Advanced Mathematical Methods for Scientists and Engineers" by Carl M. Bender and Steven A Orszag in section 6.7 Example 4. The asymptotic result is (with $x=\exp(c \beta)$) $(2 \pi)^{(1-\beta)/2} \beta^{-1/2} x^{(1-\beta)/(2\beta)}\exp(\beta x... | 5 | https://mathoverflow.net/users/37436 | 188605 | 93,394 |
https://mathoverflow.net/questions/188599 | 1 | Let $a,b,c,d\in\Bbb N$ with $c<b$.
Let $N\_+(a,b,c,d)$ be the number of monic polynomials $f\in \Bbb Z[x]$of degree $d$ with non-negative coefficients such that $$f(a)=b$$ $$f(0)=c$$
What is the value of $$\sum\_{d=0}^tN\_+(a,b,c,d)?$$
Are there sharp estimates?
| https://mathoverflow.net/users/10035 | On the number of monic polynomials | Not a complete answer, but note that by definition $N\_+(a,b,c,d)$ is the number of ways of writing $b$ as a sum $b=\sum\_{k=0}^d c\_ka^k$ with non-negative $c\_k$, and with $c\_0=c$ and $c\_d=1$. In particular it is certainly zero unless $a|(b-c)$ and $b-c\ge a^d$, in which case $N\_+(a,b,c,d):=A\big(\frac{b-c-a^d}{a}... | 10 | https://mathoverflow.net/users/6101 | 188610 | 93,395 |
https://mathoverflow.net/questions/188606 | 8 | Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically closed?
| https://mathoverflow.net/users/38889 | Adjoining torsion points from abelian varieties | If $\lambda\in\overline{\mathbb{Q}}$, the elliptic curve
$$
E\_\lambda\colon y^2=x(x-1)(x-\lambda)
$$
has $(\lambda,0)$ as $2$-torsion point and is defined over (a subfield of) $L=\mathbb{Q}(\lambda)$. Its Weil restriction $A\_\lambda:=\operatorname{Res}\_{L/\mathbb{Q}}(E\_\lambda)$ is an abelian variety defined over ... | 12 | https://mathoverflow.net/users/18238 | 188612 | 93,396 |
https://mathoverflow.net/questions/188608 | 7 | The cone ${\bf SPD}\_n({\mathbb R})$ of symmetric positive definite matrices is endowed with a nice geometrical structure. The midpoint of the (unique) geodesic between $A$ and $B$ is the so-called *geometric mean*
$$A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}.$$
>
> What is the associated distance ?
>
>
> ... | https://mathoverflow.net/users/8799 | metric on ${\bf SPD}_n({\mathbb R})$ | If I'm not mistaken, this midpoint comes from the structure of Riemannian symmetric space on the space of symmetric positive definite matrices. The Riemannian metric is defined by $g(S)(h\_1,h\_2)=trace(S^{-1}h\_2S^{-1}h\_2)$, where $S$ is in $SDP\_n$ and $h\_1,h\_2$ are symmetric matrices (tangent vectors at $S$).
T... | 7 | https://mathoverflow.net/users/8887 | 188615 | 93,397 |
https://mathoverflow.net/questions/188560 | 2 | Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function
$$g(x) = \{ g\_{jk}(x)\}\_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$
such that $0 < m\_{-}I \leq g(x) \leq m\_{+}I$, for some $m\_{-}, m\_{+} > 0$. Set $A = (A\_{1}, A\_{2}) := (-bx\_{2}/2, bx\_{1}/2)$, $b > 0$ and define the operator
$$H = \sum\_{j,... | https://mathoverflow.net/users/33804 | Geometrical interpretation of a Schrödinger operator | I'm afraid I don't know any references off the top of my head, but let me at least give you a differential-geometric translation of sorts.
1. If $g$ is real-valued, then it defines a Riemannian metric on $\mathbb{R}^2$.
2. The vector field $A$ is a vector potential for the constant magnetic field $b$, in the sense th... | 2 | https://mathoverflow.net/users/6999 | 188622 | 93,399 |
https://mathoverflow.net/questions/188616 | 2 | Let $\mathbf{Poset}$ be the category of partially ordered sets with order-preserving maps. Does $\mathbf{Poset}$ have both pushouts and pullbacks?
| https://mathoverflow.net/users/nan | Does the category of posets have pushouts and pullbacks? | Yes it is complete and cocomplete because it is locally presentable. Of course, a colimit of posets will "crush" all loops because of antisymmetry. In particular, it has pushouts and pullbacks.
| 2 | https://mathoverflow.net/users/24563 | 188624 | 93,400 |
https://mathoverflow.net/questions/188623 | 2 | Is it possible in the set theory NBG (with local choice but without global choice) that every proper class injects in the proper class of ordinals ?
| https://mathoverflow.net/users/30395 | Injection of every proper class in the ordinal class | This is equivalent to global choice, since if $V$ itself injects into Ord, then there is a global well-ordering (defined by $x<y$ if $x$ maps to a smaller ordinal than $y$), and vice versa.
| 4 | https://mathoverflow.net/users/1946 | 188625 | 93,401 |
https://mathoverflow.net/questions/188588 | -1 | A solution to $f(2x)=\alpha f(x)$ with a boundary condition $f(\beta) = \beta$ is
$$ f(x) = \left( \frac{\beta}{\alpha^{\log\_2 \beta}} \right) \alpha^{\log\_2 x}. $$
Do we know whether or not the solution is unique?
**Edit:** the function should be real-valued of a real variable, continuous, and monotonic.
| https://mathoverflow.net/users/62217 | Uniqueness of solutions of functional equations | No, it is not unique. To obtain the general solution, set $g(t)=\log f(e^t)$.
Then your equation is equivalent to $g(t+\log 2)=g(t)+\log\alpha$. One solution is an
affine function, but any periodic function can be added to it. So there are infinitely many solutions and continuity and monotonicity do not help.
| 3 | https://mathoverflow.net/users/25510 | 188627 | 93,402 |
https://mathoverflow.net/questions/186631 | 8 | Is it true (in ZFC) that for any regular infinite cardinal $\kappa$ there exists an ordered field of weight $\kappa$ and cardinality $2^\kappa$ (or at least $>\kappa$)?
The field of real numbers shows that for $\kappa=\aleph\_0$ the answer is trivially "yes".
It seems that this question has an affirmative answer in t... | https://mathoverflow.net/users/61536 | Possible cardinality and weight of an ordered field | On the one hand, if $K$ is an ordered field of weight $\kappa$, then trivially
$$|K|\le\mathrm{ded}(\kappa):=\sup\bigl\{|L|:\text{$L$ is a linear order with dense subset of size $\kappa$}\bigr\}\le2^\kappa.$$
On the other hand, we have the following lower bounds, which in particular show that there is always an OF ... | 4 | https://mathoverflow.net/users/12705 | 188628 | 93,403 |
https://mathoverflow.net/questions/188620 | 4 | Given a graph $G=(E,V)$, I construct a graph $G'$ where the vertices of $G'$ are given by the edges of $G$ and say that two edges of $G$ are neighbors in $G'$ if they have a common vertex.
Is there a standard terminology for this construction, please? Is there a book/an article with all the standard properties of the... | https://mathoverflow.net/users/62488 | What is the standard name of an edge-graph | The standard name for what you are describing is the *line graph* of $G$, however according to the *[wikipedia](http://en.wikipedia.org/wiki/Line_graph)* entry, it actually has been called the "edge graph" as well. While the wikipedia article is fairly extensive, you can also find information about line graphs in most ... | 6 | https://mathoverflow.net/users/14246 | 188632 | 93,404 |
https://mathoverflow.net/questions/188631 | 2 | Let $x=(x\_1,\ldots,x\_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant $\{x:x\_1\geq 0,\ldots,x\_n\geq0\}$?
If so, what is the polynomial of minimal degree? If not, can one find a "best" a... | https://mathoverflow.net/users/30170 | Characterizing orthants with polynomials | As stated, the answer is trivially no, since as you observe, the condition forces $p$ to also vanish on the coordinate hyperplanes, which are not contained in the positive orthant. If you modify the question to say that $p(x)>0$ iff $x$ is in the open positive orthant, the answer is still no (for $n\geq 2$). As you obs... | 3 | https://mathoverflow.net/users/75 | 188640 | 93,408 |
https://mathoverflow.net/questions/186305 | 1 | Kan extensions specify the adjoint structures between $\mathbf{Sets^{C^{op}}}$ and $\mathbf{Sets^{D^{op}}}$, where there exists a functor $f:\mathbf{C} \to \mathbf{D}$ and $\mathbf{C}$ and $\mathbf{D}$ are small categories. $(\infty,1)$-Kan extension is studied in higher topos theory, model categories, and homotopy the... | https://mathoverflow.net/users/61384 | Kan extensions and special cases | Adjoints are unique, and therefore, the functors $\Sigma\_f$ and $\Pi\_f$ are unique. As for examples of Kan extensions, take a look at the nlab page <http://ncatlab.org/nlab/show/examples+of+Kan+extensions>. I don't understand your second and third questions, but here's an example. Let $\mathbf{C}$ and $\mathbf{D}$ be... | 2 | https://mathoverflow.net/users/nan | 188645 | 93,410 |
https://mathoverflow.net/questions/188649 | 5 | Is it possible to prove in the set theory NBG (with local choice but without global choice) that the proper class of ordinals injects in every proper class ?
| https://mathoverflow.net/users/30395 | Injection of the proper class of ordinals in every proper class | The answer is no. That principle is equivalent to global choice.
To see this, consider the class $W$ consisting of all
well-orderings of any rank-initial segment $V\_\alpha$, for any
$\alpha$. If we had an injection of Ord into $W$, then there must
be unboundedly many $\alpha$'s that are used, since each
$V\_\alpha$ ... | 9 | https://mathoverflow.net/users/1946 | 188651 | 93,413 |
https://mathoverflow.net/questions/188553 | 6 | For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda\_1,\lambda\_2,\ldots\lambda\_m$ with probability distribution
$$P\_\beta(\lambda\_1,\lambda\_2,\ldots\lambda\_m)\propto \prod\_{k=1}^{m}\lambda\_k^{\beta m/2}e^{-\beta\lambda\_k/2}\prod\_{1\l... | https://mathoverflow.net/users/11260 | Quaternion Wishart matrices of half-integer dimension? | This is not an answer to trying to define a half quaternion; instead,the construction of
a random matrix model for arbitrary $a> \beta(m-1)/2$ and arbitrary $\beta> 0$ by using tri-diagonal matrices (generalizing Wishart distributions) was
done by Dumitiru and Edelman:
<http://www.math.washington.edu/~dumitriu/JMathPh... | 3 | https://mathoverflow.net/users/35520 | 188656 | 93,414 |
https://mathoverflow.net/questions/188671 | 0 | It is known that Morse-Kelly (MK) set theory forms a metatheory for ZFC. For example:
MK proves Con(ZFC). In fact, Joel David Hamkins claims in his blog post "Kelly-Morse set theory implies Con(ZFC) and much more" that in MK "there is a transitive model of ZFC, and furthermore that the universe $V$ is the union of an... | https://mathoverflow.net/users/20597 | A Question Regarding Defining Generic Extensions of ZF and ZFC in Morse-Kelly Set Theory | If there is a transitive model of ZFC, then there is a smallest ordinal $\alpha$ such that $L\_\alpha$ is a model of ZFC, and this is called the minimal transitive model of ZFC, because it is contained in all others. So we can define this model in the theory KM. Since it is countable in $L$, furthermore, there will be ... | 5 | https://mathoverflow.net/users/1946 | 188676 | 93,423 |
https://mathoverflow.net/questions/188517 | 8 | Recall that with the internal language of 1-toposes, we have the nice, basic, and useful result that geometric sequents are stable under base change along geometric morphisms: If $\varphi$ and $\psi$ are geometric formulas (formulas not containing $\forall$ and $\Rightarrow$), and if $\mathcal{E} \models \forall x:X. (... | https://mathoverflow.net/users/31233 | Base change in homotopy type theory | The answer to question 1 is yes. To see this, I think it's better to consider a "geometric sequent" to be of the form
$$ (x:X), \phi(x) \vdash \psi(x) $$
since this avoids all mention of $\forall$ and $\Rightarrow$. Now we can see that the point is just that the "geometric" logical operations $\exists$, $\wedge$, $\ve... | 2 | https://mathoverflow.net/users/49 | 188696 | 93,428 |
https://mathoverflow.net/questions/188680 | 3 | Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S = \{\boldsymbol{u} : m\_i \gt u\_i \ge 0 \} $. Alternately, you may think of $S$ as the Cartesian product of a multiple ... | https://mathoverflow.net/users/14424 | Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle | This feels awfully related to the [Subset-sum problem](https://en.wikipedia.org/wiki/Subset_sum_problem)
If $m=2(1,1,\dots,1)$, all the dot products will be the sum of subsets of components of $v$. If two such scalar product coincide, then you have solved an instance of subset-sum problem.
So, your problem is at lea... | 3 | https://mathoverflow.net/users/1056 | 188701 | 93,429 |
https://mathoverflow.net/questions/188700 | 1 | Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ and divides $\Omega$ into two regions of equal volume. I wonder if there is a numerical algorithm for constructing such $... | https://mathoverflow.net/users/42326 | Convergent algorithm for dividing a body into two regions of equal volume | There is an algorithm, based on a $\Gamma$-convergence result, which can find the minimal surface which divides a shape into two regions of equal volume. More details can be found [in this link](http://www-ljk.imag.fr/membres/Edouard.Oudet/research/Kelvin/optimal_cutting_n.php) and [this other link](http://www-ljk.imag... | 0 | https://mathoverflow.net/users/13093 | 188702 | 93,430 |
https://mathoverflow.net/questions/188707 | 14 | Is the singleton space the only Hausdorff space $X$ such that the set of automorphisms $\varphi: X\to X$ equals $\{\textrm{id}\_X\}$?
| https://mathoverflow.net/users/8628 | Hausdorff spaces with trivial automorphism group | Not at all. Those spaces are called *rigid* and there are plenty of examples in the literature. The opposite notion is *homogeneity* which is a better studied property. The first (non-trivial) rigid space was constructed by Kuratowski in "[Sur la puissance de l'ensemble des nombres de dimension au sens de M Frechet](ht... | 24 | https://mathoverflow.net/users/17836 | 188709 | 93,431 |
https://mathoverflow.net/questions/188706 | 1 | I am looking for the following paper by Cohen, F. R.:
>
> On combinatorial group theory in homotopy. Homotopy theory and its
> applications (Cocoyoc, 1993), 57–63, Contemp. Math., 188, Amer. Math.
> Soc., Providence, RI, 1995.
>
>
>
Many thanks in advance for finding the paper for me!
| https://mathoverflow.net/users/15770 | Request a paper by Fred Cohen | [You can find here.](http://www.ams.org/books/conm/188/conm188.pdf)
In case if the link does not work, you can give your e-mail id. I will send you via e-mail.
| 1 | https://mathoverflow.net/users/26631 | 188718 | 93,433 |
https://mathoverflow.net/questions/188716 | 4 | (Hope this question is o.k. for MO)
I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular Fourier multiplier); to solve it.
But mainly with underlying framework function spaces, which I have seen so far, are functi... | https://mathoverflow.net/users/33018 | PDEs on torus $\mathbb T$ | It is a bit old, but I would check [Bourgain](http://press.princeton.edu/chapters/s8447.pdf) first.
A more recent reference is (among many others) the preprint of [Strunk](https://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb14006.pdf).
| 3 | https://mathoverflow.net/users/12898 | 188719 | 93,434 |
https://mathoverflow.net/questions/188729 | 13 | A space $(X,\tau)$ is called *rigid* if $\textrm{Aut}(X)=\{\textrm{id}\_X\}$. We say $(X,\tau)$ is *strongly rigid* if for every continuous map $f:X\to X$ we have that $f = \textrm{id}\_X$ or $f$ is constant (that is there is $x\_0\in X$ such that $f(x)=x\_0$ for all $x\in X$).
Is there a strongly rigid Hausdorff spa... | https://mathoverflow.net/users/8628 | Strongly rigid Hausdorff spaces | Yes, see for example "[Continua which admit only the identity mapping onto non-degenerate subcontinua](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv60i1p23bwm)" by H. Cook (Fund. Math. 60, 1967, 241-249).
| 17 | https://mathoverflow.net/users/17836 | 188735 | 93,440 |
https://mathoverflow.net/questions/157458 | 1 | let the entries of a rectangular matrix $A\in\mathbb{C}^{m\times n}; m,n\in\mathbb{N}$ be stored in row-order in a linear vector $v$, i.e. $A\_{i,j}=v\_{i\*m+j}$
**Question:**
How can the first element and the size of the next cycle in the permutation $\mathcal{P}(v\_i)$ of the entries of $v$ be calculated, when ... | https://mathoverflow.net/users/31310 | Cycles of Permutation Related to Rectangular Matrix Transposition | I just found the paper "**In-Place Transposition of Rectangular Matrices**" by **Fred G. Gustavson and Tadeusz Swirszcz**, which provides a solution for the problem.
An online version of the paper can be found here:
<http://www.orcca.on.ca/conferences/cca2008/papers/gustavson.pdf>
| 1 | https://mathoverflow.net/users/31310 | 188747 | 93,443 |
https://mathoverflow.net/questions/188726 | 2 | Consider a 2-player symmetric game given by a payoff matrix $A\in [0,1]^{n,n}$ for the row player (i.e. the column player matrix is $A^t$).
Let $s=<s\_1,s\_2>$ be a (possibly mixed-strategies, not-necessarily symmetric) equilibrium for the game.
Define $Sup(s)=Sup(s\_1)\cup Sup(s\_2)$, where $Sup(s\_i)$ is the set ... | https://mathoverflow.net/users/47499 | Is there always a symmetric "subset equilibrium" for an equilibrium in a symmetric game? | Here is a counterexample (from p.76 of [my thesis](http://www.mit.edu/~nstein/documents/DoctoralThesis.pdf)):
$$
A = A^T = \begin{bmatrix}
0 & 3 & 2 \\ 3 & 0 & 2 \\ 2 & 2 & 3
\end{bmatrix}.
$$
Labeling the strategies in order as $a$, $b$, and $c$, there are asymmetric Nash equilibria $(a,b)$ and $(b,a)$ with suppor... | 3 | https://mathoverflow.net/users/5963 | 188749 | 93,444 |
https://mathoverflow.net/questions/188759 | 6 | [Wikipedia](http://en.wikipedia.org/wiki/Symmetric_monoidal_category) states:
>
> The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an infinite loop space.
>
>
>
If my mind is correct, Segals delooping machine gives a functor $Sp$ from topological symmetric monoidal... | https://mathoverflow.net/users/62554 | Is the classifying space of a symmetric monoidal category an infinite loop space? | You need group Completion, indeed.
| 5 | https://mathoverflow.net/users/21985 | 188760 | 93,447 |
https://mathoverflow.net/questions/188755 | 4 | Suppose to have two real vector spaces $V$ and $W$ and an injective map $T:V\rightarrow W$ such that $T(\alpha v)=\alpha T(v)$ for all $v\in V$ and $\alpha \in\mathbb{R}$. Do there exist some conditions on the vector spaces or the map which guarantee that $T$ is also additive, that is $T(v+u)=T(v)+T(u)$?
| https://mathoverflow.net/users/43915 | When a homogeneous map between vector spaces is also additive? | If $V$ is one dimensional, every homogeneous map $T:V\to W$ is necessarily linear and thus additive.
If $W$ is one dimensional, it follows from the injectivity assumption on $T$ that $\dim(V)\leq1$ and so every homogeneous map is additive.
Let us then assume that both spaces have dimension two or higher but finite.
I... | 6 | https://mathoverflow.net/users/55893 | 188765 | 93,450 |
https://mathoverflow.net/questions/188728 | 0 | Could anyone provide some hints for solving:
$\underset{n} {\mathrm{argmin}} \frac{a}{r + ns} + \sum\_{i=0}^{n-1}\frac{b}{r + is}$ for $n \in \{1,2,3,\ldots\}$
The problem is part of a coding exercise, but I was curious whether an analytic solution could be derived.
| https://mathoverflow.net/users/62534 | Analytic solution $\underset{n} {\mathrm{argmin}} \frac{a}{r + ns} + \sum_{i=0}^{n-1}\frac{b}{r + is}$ | Let
$$ F(n) = \dfrac{a}{r+ns} + \sum\_{i=0}^{n-1} \dfrac{b}{r+is}$$
I'll assume $a,b,r,s > 0$. Note that
$$F(n+1) - F(n) = \dfrac{bs n + b r+ bs - as}{(ns + r + s)(ns + r)}$$
So you want the least positive integer $n$ (if any) such that
$b s n + br + bs - as \ge 0$, i.e. $n = \left\lceil \dfrac{a}{b} - \dfrac{r}{s} - 1... | 2 | https://mathoverflow.net/users/13650 | 188770 | 93,452 |
https://mathoverflow.net/questions/188775 | 3 | I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map
$$Top(T,P) \rightarrow Top (T,A) \times\_{Top(T,B)} Top (T,C)$$
is a bijection of sets. Now both sides are equipped with a topology, so I'm won... | https://mathoverflow.net/users/26470 | Is the defining bijection for a pullback of topological spaces a homeomorphism? | Yes. The functor $Top(T,-)$ preserves limits because it is a right adjoint.
| 13 | https://mathoverflow.net/users/6666 | 188776 | 93,453 |
https://mathoverflow.net/questions/188721 | 5 | It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty\_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ for $k=0, 1, 2$.
My question: Is this an if and only if? That is, if $C^\infty\_c(M)$ is dense in these Sobolev spa... | https://mathoverflow.net/users/16702 | Compactly supported functions and Sobolev spaces on manifolds | The general answer to this question is no. Global bound on the Ricci curvature is not necessary for the density of smooth functions with compact supports.
Indeed, when $(M,g)$ is a smooth complete Riemannian manifold with positive injectivity radius and lower bound for the Ricci curvature, then the smooth functions w... | 3 | https://mathoverflow.net/users/26608 | 188779 | 93,455 |
https://mathoverflow.net/questions/188772 | 3 | Can one, for an infinite set A in ZFC, use forcing to add so many generic subsets of A as to make the collection of all subsets of A a proper class? Consider now a model $M$ of ZFC and use $Add( , )$ to make the collection of all subsets of any given infinite set in the forcing extension $M$[$G$] a proper class. This f... | https://mathoverflow.net/users/20597 | A question regarding forcing extensions | $\newcommand\Ord{\text{Ord}}
\newcommand\Add{\text{Add}}$Yes, one may undertake such kind of class forcing constructions.
For example, we can force with $\Add(\omega,\Ord)$ to add $\Ord$
many Cohen reals. As you point out, we won't have ZFC in the
resulting forcing extension, but we will get a sensible model of
some th... | 5 | https://mathoverflow.net/users/1946 | 188781 | 93,456 |
https://mathoverflow.net/questions/188677 | 3 | Is there an explicit formula for the following quantity?
$$f\_m(a\_1,\ldots,a\_n):=\sum\_{\substack{k\_1+\ldots+k\_n=m \\ k\_1,\ldots,k\_n\in \mathbb{N}}} k\_1^{a\_1}\ldots k\_n^{a\_n}\ ,\hspace{1cm} m,a\_1,\ldots,a\_n\in \mathbb{N}$$
(for instance $f\_m(0,\ldots,0)$ is simply the number of compositions of m into n... | https://mathoverflow.net/users/27069 | Sum over integer compositions of $m$ with $n$ parts of a fixed monomial in the parts | I'm not sure if there is anything simpler than $f\_m(a\_1,\dots,a\_n)$ being the coefficient of $x^m$ in the product of [polylogarithms](http://en.wikipedia.org/wiki/Polylogarithm):
$$\mathrm{Li}\_{-a\_1}(x)\cdot \mathrm{Li}\_{-a\_2}(x)\cdots \mathrm{Li}\_{-a\_n}(x).$$
In particular, since $\mathrm{Li}\_{-1}(x) = x\c... | 3 | https://mathoverflow.net/users/7076 | 188782 | 93,457 |
https://mathoverflow.net/questions/188785 | 2 | Let $A\rightarrow B$ be a commutative $A$-algebra. If $A$ is a field and $B$ Noetherian and formally smooth over $A$, then it is known that $B$ must be a regular ring. Is there a partial converse of this result, asserting that "$B$ is a regular ring + some further conditions on $A$ and $B$ $\implies$ $B$ is formally sm... | https://mathoverflow.net/users/62558 | Regular rings and formally smooth algebras | Here is EGA IV$\_1$, Chap. 0, Th. 22.5.8:
>
> Let $k$ be a field with characteristic exponent $p$, and let $A$ be a local Noetherian $k$-algebra. Then the following are equivalent:
>
>
> 1. $A$ is a formally smooth $k$-algebra for its preadic topology,
> 2. $A$ is geometrically regular over $k$,
> 3. for all fini... | 5 | https://mathoverflow.net/users/17988 | 188789 | 93,459 |
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