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/142086 | 16 | Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size $\kappa$.
**Edit:**
In response to some attempts at a positive answer, let me explain something that does not work. If $... | https://mathoverflow.net/users/11145 | Singularizing forcing of "small" cardinality? | The answer to your question is no. We have the following theorem.
>
> **Theorem.** Suppose $\kappa$ is a regular uncountable cardinal and $|P|=\kappa.$ Then $\Vdash\_P cf(\kappa)=|\kappa|.$
>
>
>
**Proof.** Let $\tau$ be a name of an unbounded subset of $\kappa.$ We show there is $f\in V, f:\kappa\to\kappa$ s... | 11 | https://mathoverflow.net/users/11115 | 196742 | 95,727 |
https://mathoverflow.net/questions/196739 | 7 | The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the loop of $G$ with the 'positive energy' condition. I'm trying to find a concise mathematical definition of the product on... | https://mathoverflow.net/users/4177 | Concise mathematical definition of the fusion product on the Verlinde ring? | The fusion product arises from the braided monoidal structure on the modular tensor category of [adjectives] loop group representations at level $k$ itself. This is the value $Z(S^1)$ of Chern-Simons theory with gauge group $G$ and level $k$ on the circle, and accordingly its braided monoidal structure comes from think... | 5 | https://mathoverflow.net/users/290 | 196743 | 95,728 |
https://mathoverflow.net/questions/196735 | 4 | In [this paper](http://arxiv.org/abs/math/9605234), Gitik and Shelah make the following claim (part of Proposition 1.5):
>
> **Claim (Gitik-Shelah):** Suppose $\kappa < \lambda$ are regular, $2^\lambda = \lambda^+$, and $D$ is a normal ideal on $\lambda$. If forcing with $D^+$ makes $cf(\lambda^+) < \kappa$, then i... | https://mathoverflow.net/users/11145 | Presaturated ideals | The result follows from the following theorem:
>
> **Theorem.** Suppose $\kappa$ is a regular uncountable cardinal and $|P|\leq \kappa.$ Then $\Vdash\_P cf(\kappa)=|\kappa|.$
>
>
>
In your case $D^+$ has size $\lambda^+$ and it forces $cf(\lambda^+)<\kappa.$ So by above theorem it also forces $|\lambda^+|<\kap... | 4 | https://mathoverflow.net/users/11115 | 196744 | 95,729 |
https://mathoverflow.net/questions/130447 | 8 | Let $\lambda$ be a singular cardinal. Is it consistent that there is a forcing of size $\lambda^+$ that collapses $\lambda^+$ while preserving all cardinals below $\lambda$?
(Note that even without the size requirement this implies a failure of the Jensen covering property, so such a forcing does not necessarily exis... | https://mathoverflow.net/users/11145 | collapsing successor of singular | The answer is no and it follows easily from the following theorem:
>
> **Theorem.** Suppose $\kappa$ is a regular uncountable cardinal and $|P|\leq \kappa.$ Then $\Vdash\_P cf(\kappa)=|\kappa|.$
>
>
>
For a proof of the theorem see
[Singularizing forcing of "small" cardinality?](https://mathoverflow.net/questi... | 5 | https://mathoverflow.net/users/11115 | 196745 | 95,730 |
https://mathoverflow.net/questions/196717 | 3 | For $n := (n\_1,\dots,n\_N) \in \mathbb{N}\_{>1}^N$, let $X\_n := \prod\_{j=1}^N [n\_j]$, where as usual $[m] := \{1,\dots,m\}$.
>
> Are there any known generic constructions for (Hamming) sphere packings in $X\_n$ other than the "trivial" ones that essentially embed each factor $[n\_j]$ in some $\mathbb{F}\_{p^{r... | https://mathoverflow.net/users/1847 | Block error-correcting codes over inhomogeneous alphabets | I think what you are looking for is **mixed codes**.
A good start point would be [Brouwer--Hämäläinen--Östergård--Sloane](http://www.ams.org/mathscinet-getitem?mr=1486654). They are talking about mixed binary/ternary code, so for some $k$, $n\_1=\cdots=n\_k=2$ while $n\_{k+1}=\cdots=n\_N=3$. Brouwer keep an [online l... | 2 | https://mathoverflow.net/users/20595 | 196746 | 95,731 |
https://mathoverflow.net/questions/196697 | 1 | Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional derivative
$$
D^\alpha f(x)=C\_\alpha \frac{d}{dx}\int\_0^x(x-y)^{-\alpha}f(y)\,dy,\quad \alpha\in(0,1),
$$
where $f\in \Lambda... | https://mathoverflow.net/users/14551 | What is the fractional derivative smoothness of functions from the Zygmund class? | It is indeed what you expect. A rather straightforward proof of this fact when $[0,1]$ is replaced with $\mathbb{R}^n$ relies on Littlewood-Paley decomposition and is detailed for instance in [this book](http://ukcatalogue.oup.com/product/9780198503972.do). Since your definition of $\Lambda$ is only local, you may simp... | 2 | https://mathoverflow.net/users/62629 | 196750 | 95,734 |
https://mathoverflow.net/questions/196741 | 1 | Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$.
The *Ricci curvature* can be viewed as a differential operator $\text{Rc}:\Gamma(S^2\_+M)\rightarrow\Gamma(S^2M)$ from the space of metrics on $M$ to the space of symmetric tensors on $M$ given by $g\mapsto\text{Rc}(g)$.
... | https://mathoverflow.net/users/43445 | Linearisation of Einstein operator | Let $\Phi\_s$ denote the one parameter family of diffeomorphisms generated by $X$ and let $g(s) = \Phi\_s^\* g$. Note that $r(\Phi\_s^\* g) = \Phi\_s^\* r(g)$ by definition, so you have that the right hand side of your equation is exactly
$$ L\_X r(g) = \lim\_{s\to 0} \frac{\Phi\_s^\* r(g) - r(g)}{s} .$$
On the othe... | 5 | https://mathoverflow.net/users/3948 | 196751 | 95,735 |
https://mathoverflow.net/questions/196752 | 2 | An *edge clique cover* of an undirected graph $G$ is a set of cliques of $G$ such that every edge of $G$ is an edge in at least one clique in the set. The *edge clique cover number* $\theta(G)$ is the minimum number of cliques in an edge clique cover of $G$. Equivalently, this is the [intersection number](http://en.wik... | https://mathoverflow.net/users/39475 | Upper bounds on the edge clique cover number on special graph classes | The 1995 paper "A Survey of Clique and Biclique Coverings and Factorizations of (0,1)-Matrices" by Sylvia D. Monson, Norman J. Pullman and Rolf Rees in Bulletin of the ICA, Vulume 14, 17-86, might contain results related to your question. If you cannot find the paper, please send me a note and I can email you a scanned... | 4 | https://mathoverflow.net/users/62709 | 196770 | 95,742 |
https://mathoverflow.net/questions/196753 | 14 | For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim\_n \operatorname{Sp}\_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary matrices, let $\operatorname{BSp}(R)$ be a classifying space, let $\operatorname{BSp}(R)^+$ be the result of applying Quill... | https://mathoverflow.net/users/5339 | Symplectic K-theory | **Concerning the definition of symplectic K-theory:** there are various possible definitions, the homotopy groups of the plus-construction of the classifying space of the infinite symplectic group is one such definition. Other definitions can be given using categories of forms (similar to Q-construction or $S^{-1}S$-co... | 19 | https://mathoverflow.net/users/50846 | 197767 | 95,744 |
https://mathoverflow.net/questions/196733 | 4 | In the abstract of [this paper](http://65.54.113.26/Publication/16030379/atoms-and-antiatoms-in-the-lattice-of-group-topologies), it is said that a minimal group topology on an abelian group is not Hausdorff.
Suppose $G$ is an abelian group and $\mathcal T$ is a minimal group topology on $G$ and let $N$ be the inters... | https://mathoverflow.net/users/47958 | Can an abelian group have a minimal group topology? | The paper that you mentioned is referring to a theorem proved by D. Remus in his dissertation:
>
> **Theorem:** If $G$ is abelian and $\tau$ is a minimal group topology on $G$, then there exists a subgroup $N$ such that $G/N \cong \mathbb{Z}/p\mathbb{Z}$ ($p$ prime) and $\{N\}$ is a fundamental system of neighborho... | 3 | https://mathoverflow.net/users/17836 | 197770 | 95,745 |
https://mathoverflow.net/questions/197773 | 1 | Let $S\subset M\_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
>
> What matrices belongs to $S$, precisely?
>
>
>
Let $M=Det^{-1}\{0\}-S$ be the codimension one submanifold of $M\_{n}(\mathbb{R})$ which has a natural Riemannian ... | https://mathoverflow.net/users/36688 | A geometric property of singular matrices | Your first question has an easy answer. The differential of $\det$ is
$$\sum\_{i,j}\hat a\_{ij}{\rm d}a\_{ij},$$
where $\hat A$ is the cofactor matrix. Thus a singular point is such that $\hat A=0\_n$, in other words, it has rank $\le n-2$.
Actually, ${\bf M}\_n({\mathbb R})$ can be stratified by the sets $R\_0,\ldot... | 7 | https://mathoverflow.net/users/8799 | 197775 | 95,747 |
https://mathoverflow.net/questions/47172 | 10 | Let $\kappa$ be a regular cardinal. If I understand correctly, the proof that the intersection of $<\kappa$ many club subsets of $\kappa$ is a club does not require AC. However, the proof that the club filter on $\kappa$ is $\kappa$-complete does ostensibly require AC because given a sequence of sets in the club filter... | https://mathoverflow.net/users/3183 | Completeness of the club filter without AC | If $M$ is an inner model of a forcing extension $V[G]$ of a model $V$ of Choice via a partial order of cardinality $\kappa$ in $V$, then for every regular cardinal $\lambda$ of $V$ greater than $\kappa$, every club subset of $\lambda$ in $V[G]$ will contain one from $V$. So, in $M$, the club filter on $\lambda$ will be... | 6 | https://mathoverflow.net/users/31807 | 197779 | 95,748 |
https://mathoverflow.net/questions/197781 | 5 | The question started from a problem brought home by a friend's 5th grader: "How many ways can you seat 5 people around a round table so that the people sitting to the left of any person is different in each seating arrangement?"
Here is a snap solution:
For seating arrangements of $N$ people consider a directed grap... | https://mathoverflow.net/users/38448 | How many hamiltonian cycles can be removed from a complete directed graph before it becomes disconnected? | I will rephrase your question slightly. Let $K\_{n}^{\*}$ be the directed graph with $n$ vertices and two oppositely directed edges for each pair of vertices. Your question is then the following.
>
> What is the maximum number of edge-disjoint directed Hamiltonian cycles of $K\_{n}^{\*}$?
>
>
>
For $n=2k+1$ od... | 10 | https://mathoverflow.net/users/2233 | 197783 | 95,749 |
https://mathoverflow.net/questions/164125 | 6 | Let $A\_1,A\_2,A\_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e\_1,e\_2,e\_3\}$ is the standard basis in $\mathbb{R}^3$, we can construct a polyhedron
$$ P = \mathrm{conv}(A\_1,A\_2,A\_3) + \mathrm{cone}(e\_1,e\_2,e\_3) $$
where $\mathrm{conv}$ d... | https://mathoverflow.net/users/15092 | Triangulations of special polyhedra | Unless you have some additional structure on the points $A\_1$,...,$A\_n$ your problem is as hard (and as easy) as computing the convex hull of $2n$ points in $\mathbb{R}^n$, and triangulating this convex hull.
EDIT:
Neither the fact that you have zeroes nor the value of the determinant simplify the problem, as fa... | 4 | https://mathoverflow.net/users/22608 | 197790 | 95,751 |
https://mathoverflow.net/questions/197788 | 1 | let $\Omega\subset M$ be an open and unbounded set in a smooth manifold $M$ with boundary $\partial \Omega$. Now let $p\_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\Omega$ which vanishes on the boundary $\partial \Omega$, i.e. for all $y\in \Omega$:
$(i) (\partial\_t -\Delta\_x )p\_t(x,y)... | https://mathoverflow.net/users/21870 | Fundamental solution to the heat equation with zero boundary values | Without imposing some growth condition at infinity, this is not to be expected. There are nonzero solution of the heat equation on the real line which satisfy a zero initial condition.
| 2 | https://mathoverflow.net/users/12120 | 197791 | 95,752 |
https://mathoverflow.net/questions/197784 | 4 | The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping from its beta-Cpt to the range space (Engelking, Outline, Chapter 5.3). For these reasons, it has performed well as the "un... | https://mathoverflow.net/users/66946 | How is a MacNeille completion "universal" like a beta-compactification is "universal"? | A subset $A$ of a complete lattice $L$ is said to be join-dense if $L=\{\bigvee^{L} R\mid R\subseteq A\}$ and $A$ is said to be meet-dense in $L$ if $L=\{\bigwedge^{L}R\mid R\subseteq A\}$. It turns out that the Dedekind–MacNeille completion of a poset $P$ is up to an isomorphism preserving $P$ the only complete lattic... | 5 | https://mathoverflow.net/users/22277 | 197803 | 95,757 |
https://mathoverflow.net/questions/160507 | 12 | When S is a subset of an inner product space, let d(S) denote ${\sum\limits\_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, and V is a subspace of $\mathbb{R^n}$.
Does this hold in general: ${\frac{d(L)} {d(L \cap V)}} \geq {\frac{d(M)}{d(M \... | https://mathoverflow.net/users/7935 | Inequality regarding sum of gaussian on lattices | Oded Regev and I just posted a [paper](http://arxiv.org/abs/1502.04796) to the arXiv that answers this question. See Corollary 4.4. In particular, we show that the OP's inequality does in fact hold and that it follows from a new inequality concerning the Gaussian mass of lattice cosets.
Interestingly, we came to thi... | 11 | https://mathoverflow.net/users/64613 | 197812 | 95,763 |
https://mathoverflow.net/questions/186428 | 15 | Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the norm according to h).
Suppose $H\_g$ and $H\_h$ are the heat kernels on (M, g) and (M, h) respectively.
Does ${\frac{H\_g(x... | https://mathoverflow.net/users/7935 | Is the heat kernel more spread out with a smaller metric? | Oded Regev and I just posted a [paper](http://arxiv.org/abs/1502.04796) to the arXiv that shows that this does in fact hold over $\mathbb{R}^n/\mathcal{L}$ where $\mathcal{L}$ is a lattice. See Proposition 4.2.
The general case does not hold. In particular, if the second-largest eigenvalue of the heat kernel is uniq... | 9 | https://mathoverflow.net/users/64613 | 197813 | 95,764 |
https://mathoverflow.net/questions/197818 | 1 | Let $f:[0,2\pi]\rightarrow R^2$ be a smooth function such that $f([0,2\pi])$ is a smooth closed simple curve $C$. Suppose $(0,0)$ lies inside the the bounded open region enclosed by $C$ and $f(t)=(x(t),y(t))$. Is it true that $g(t)=x^2(t)+y^2(t)$ has at least 4 critical points in $[0,2\pi)$?
| https://mathoverflow.net/users/42326 | Number of critical points | My original answer had a silly mistake in it.
*Updated Answer:* There exists a smooth closed simple curve, which encloses the origin, yet whose squared distance to the origin has only 2 critical points in $[0,2\pi)$.
Let $f(t)=(\frac{1}{2}+\cos{t}, \sin{t})$. This is simply the unit circle shifted to the right by ... | 6 | https://mathoverflow.net/users/68222 | 197820 | 95,765 |
https://mathoverflow.net/questions/197814 | 3 | Let $f(x)=\frac{1}{\sin(\pi x)}$ for $x\in (0, 1)$ and let
$\Gamma=\left\{(x,f(x)): x\in (0, 1)\subset \mathbb{R}^2\right\}$ be its graph.
For any set $X\subset \mathbb{R}^2$ and $\lambda>0$ and $\mathbf{v}\in \mathbb{R}^2$, let $\lambda X+\mathbf{v}=\left\{\lambda \mathbf{x}+\mathbf{v}: \mathbf{x}\in X\right\}$.
L... | https://mathoverflow.net/users/66607 | Is this a $C^0$ foliation of $\mathbb{R}^2$? | No, your partition of $\mathbb R^2$ into curves is not a foliation in the sense you provide. To see this, you only have to notice that any neighborhood $U$ of $(0 , y\_0)$ disconnects some curves and does not others. Being given a rectifying chart $(\psi , U)$ and a strict subdomain $U'\subset U$ with smooth boundary, ... | 5 | https://mathoverflow.net/users/24309 | 197826 | 95,767 |
https://mathoverflow.net/questions/196759 | 10 | Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory?
By "interesting/useful", I mean "not merely descriptive", that is, they can actually say something new and not just formalize well known concepts.
For example, I have r... | https://mathoverflow.net/users/30366 | Applications of cohomology to probability and statistics | There is a very nice interpretation of entropy as a cohomology class by Baudot and Bennequin which you can read about [HERE](http://djafari.free.fr/MaxEnt2014/papers/97_paper.pdf).
In general, I strongly believe that there is an underlying topological content to parts of information theory- as there is information ge... | 11 | https://mathoverflow.net/users/2051 | 197830 | 95,770 |
https://mathoverflow.net/questions/197808 | 6 | A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian line bundle $L\to M$ whose Chern class satisfies $c\_{1}(L) = [\omega]$. I have the following questions:
* Does anyone k... | https://mathoverflow.net/users/66688 | Line bundles over Kähler–Hodge manifolds | At least when $M$ is compact, which I assume here,
the standard reference for this subject is probably the book
* *Algebraic Geometry*, Griffiths & Harris
but the following books are excellent references as well :
* *Differential analysis on complex manifolds*, Wells
* *[Complex Analytic and Differential Geometry... | 6 | https://mathoverflow.net/users/10696 | 197831 | 95,771 |
https://mathoverflow.net/questions/197816 | 2 | Let $D$ be the unit disk in $\mathbb{R}^{n}$, we consider the $n$ dimension wave equation defined on $D$,
$$\square u=F$$
where $\square=\partial\_{t}^{2}-\triangle$ is the standard wave operator in $\mathbb{R}^{1+n}$, together with the initial and boundary conditions
$$u(0,x)=u\_{0},u\_{t}(0,x)=u\_{1}, u(t,x)|\_{\... | https://mathoverflow.net/users/67007 | Boundary energy estimate of wave equations | This is a classical but intricate question : what are the boundary and interior estimates for a hyperbolic initial-boundary value problem ? The answer is clean for the wave equation, because the boundary is non-characteristic. You should consider $L^2$-spaces in both space and time variables. A typical estimate is
$$\g... | 3 | https://mathoverflow.net/users/8799 | 197832 | 95,772 |
https://mathoverflow.net/questions/196192 | 2 | Theorem 1.67 On page 19 of Ken Ono's book The Web of Modularity says:
Every modular form on $SL\_2(\mathbb{Z})$ may be expressed as a rational function in $\eta(z)$, $\eta(2z)$ and $\eta(4z)$.
The proof uses the following two facts,
1, As graded algebra, $M(SL\_2(\mathbb{Z}))$ is generated by $E\_4(z)$ and $E\_6(... | https://mathoverflow.net/users/31134 | Relations of eisenstein series with eta quotient | This does not answer your question, but I want to elaborate on my comment to Jeremy's answer that such identities can also be understood from the viewpoint of elliptic functions. I use as a reference Vol. 2 of Erdelyi et al., Higher Transcendental functions. By Eq. (13.20.8),
$$g\_2=C\left(\theta\_2^8+\theta\_3^8+\thet... | 5 | https://mathoverflow.net/users/10846 | 197836 | 95,773 |
https://mathoverflow.net/questions/197841 | 2 | I'm looking at a family $(f\_t)$ of densities of some continuous random variables and know that
$$\int\_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f\_t(\xi) \mathrm{d} \xi \xrightarrow{t \to \infty} \Phi(u)$$
for any $u \in \mathbb{R}$, where $\Phi$ denotes the PDF of a standard Gaussian random variable. Does... | https://mathoverflow.net/users/57022 | Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$? | A sketch of an argument (I'll leave it to you to figure out the precise conditions and formulations that makes this argument work):
Since $\Phi$ is [even, let us just focus on the problem restricted to $\mathbb{R}\_+$](https://mathoverflow.net/questions/197841/does-int-phi-left-fracu-xi-right-f-t-xi-mathrmd-xi-to-phi... | 2 | https://mathoverflow.net/users/3948 | 197845 | 95,778 |
https://mathoverflow.net/questions/197848 | 3 | Let $E \subset \mathbb{P}^3\_{\mathbb{R}}$ be a real elliptic normal curve with two non-null-homotopic connected components. Is there a parametrization
$$ \chi: (\mathbb{R}/\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z}) \longrightarrow E$$
such that any four points $P\_1,P\_2,P\_3,P\_4$ on $E$ are coplanar if and only if t... | https://mathoverflow.net/users/1792 | How can one parametrize a real elliptic normal curve such that four points are coplanar iff their parameters sum to zero? | It looks like "no", even if you do not assume any special kind of regularity. The image of $0$ would have to be a point of $4$-fold intersection with a plane (some kind of maximal "flex"). On the other hand, since each component is not null-homologous, each plane intersects **each** component, hence, there's no **real*... | 7 | https://mathoverflow.net/users/44953 | 197850 | 95,779 |
https://mathoverflow.net/questions/194430 | 12 | (Duplicated from math.stackexchange)
I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given [here](https://math.stackexchange.com/questions/1112419/intuitive-approach-to-de-rham-cohomology). It is not enough to know ... | https://mathoverflow.net/users/62367 | Intuitive Aproach to Dolbeault Cohomology | Here is some nonsense that I find useful: On a complex manifold,
$$\frac{\mbox{locally constant functions}}{\mbox{smooth functions}} \approx \frac{\mbox{locally constant functions}}{\mbox{holomorphic functions}} \cdot \frac{\mbox{holomorphic functions}}{\mbox{smooth functions}}$$
The left hand side is where good ge... | 17 | https://mathoverflow.net/users/297 | 197852 | 95,780 |
https://mathoverflow.net/questions/196109 | 6 | More generally, I expect that the following is true:
>
> Let $D$ be a diagram quasicategory, let $d \in D$ be a vertex, and use this to define $D' = D \amalg\_{\Delta^{\{0\}}} \Delta^1$. Then $\mathrm{Fun}(D',C) \to \mathrm{Fun}(\Delta^{\{1\}},C) \cong C$ is a cocartesian fibration.
>
>
>
(This follows from Co... | https://mathoverflow.net/users/303 | For a quasicategory $C$, why is $\mathrm{Fun}(\Lambda^2_0,C) \to \mathrm{Fun}(\Delta^{\{2\}},C) \cong C$ a cocartesian fibration? | This follows from (the coCartesian version of) HTT Corollary 2.4.7.12, applied to the "evaluation at $d$" functor $\mathrm{Fun}(\mathcal{D}, \mathcal{C}) \rightarrow \mathcal{C}$, since we can identify $\mathrm{Fun}(\mathcal{D}', \mathcal{C})$ with $\mathrm{Fun}(\mathcal{D}, \mathcal{C}) \times\_{\mathcal{C}} \mathrm{F... | 5 | https://mathoverflow.net/users/1100 | 197861 | 95,782 |
https://mathoverflow.net/questions/197862 | 3 | Kurt Gödel in 1931 used $x\Pi a$ where we in contemporary notation would use $(\forall x) A$ or $(x)A$, and $Ex a$ where we would use $(\exists x) A$. I believe that I remember that $\Sigma xA$ has been used with the meaning $\exists x A$. Is my belief correct, and if so by what authors was $\Sigma x$ so used? What, if... | https://mathoverflow.net/users/37385 | Was $\Sigma x$ used as quantifier? | According to [Wikipedia](http://en.wikipedia.org/wiki/Quantifier_%28logic%29),
* Charles Sanders Peirce used $\Pi\_x$, $\Sigma\_x$ in 1885;
* Guiseppe Peano used (x), $(\exists x)$ in 1897;
* Gentzen introduced $\forall x$ in 1935;
* $(\forall x)$, $(\exists x)$ became standard in the 1960s.
| 4 | https://mathoverflow.net/users/4600 | 197864 | 95,783 |
https://mathoverflow.net/questions/197817 | 3 | Let $X$ be a normal variety over $\mathbb{C}$ and $\pi:\tilde{X}\rightarrow X$ a log resolution with (reduced) exceptional divisor $E$. Let $U$ be the smooth locus of $X$ and $\omega$ a holomorphic 1-form on $U$, when is it possible to extend $\omega$ to a holomorphic form on $\tilde{X}$ with at most logarithmic poles ... | https://mathoverflow.net/users/51119 | Extending holomorphic forms | Here is a result which is sort of what you are asking:
>
> **Theorem** (Greb-Kebekus-Kovács-Peternell)
> Let $X$ be a complex quasi-projective variety of dimension $n$
> and let $D$ be a $\mathbb Q$-divisor on $X$ such that the pair
> $(X, D)$ is log canonical. Let $\pi:\widetilde X\to X$ be a log resolution
> ... | 4 | https://mathoverflow.net/users/10076 | 197866 | 95,784 |
https://mathoverflow.net/questions/197874 | 16 | I am trying to understand the statement that a Deligne-Mumford stack is locally a quotient $[U/G]$, where $G$ is a finite group. I don't understand why you can make $G$ a finite group, instead of a general group scheme. For example, if $G$ is a nonconstant étale group scheme over $S$, and $\mathcal{X}=[S/G]$ a quotient... | https://mathoverflow.net/users/38276 | The difference between an étale finite group scheme and a finite group | The way I think about finite etale group schemes over a connected scheme S, is that it's a group object in the category of schemes finite etale over S. This latter category is well-known to be equivalent to the category of finite sets X together with a action of $\pi\_1(S,s)$ (pick some geometric point $s\in S$). This ... | 30 | https://mathoverflow.net/users/15242 | 197876 | 95,787 |
https://mathoverflow.net/questions/197877 | 2 | According to answer of Denis Serre to [this question](https://mathoverflow.net/questions/197773/a-geometric-property-of-singular-matrices), the manifold of singular matrices in $M\_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M\_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$
So we define a (line bundle) over this m... | https://mathoverflow.net/users/36688 | A line bundle over the manifold of singular matrices | No, this bundle is not trivial (starting from dimension $2$). Introduce a metric, consider projector to a hyperplane, and rotate this hyperplane through $\pi$ about an axis. You get an orientation reversing loop.
| 3 | https://mathoverflow.net/users/44953 | 197879 | 95,788 |
https://mathoverflow.net/questions/197871 | 6 | I'm trying to take familiarity with homotopy theory and I have the following questions. Let $\mathcal{C}$ be a small category, and let $F\: : \:\mathcal{C}\to \mathcal{M}$ that take values in a (simplicial) model category.
For any simplicial presheaves $K\: : \: \mathcal{C}^{op}\to sSet$, I denote with
$$
K\otimes\_{\m... | https://mathoverflow.net/users/41970 | homotopy tensor product of functors and bar construction | Yes. (Of course for these constructions to be homotopically well-behaved you need $F$ to be levelwise cofibrant.) In fact, assuming that such $QK$ exists we can express it by an explicit formula which then proves that it indeed exists. All we need to know is that tensoring over $\newcommand{\C}{\mathcal{C}}\C$ with a f... | 5 | https://mathoverflow.net/users/12547 | 197888 | 95,790 |
https://mathoverflow.net/questions/197904 | 4 | Disclaimer: This is a cross-listing of a [math.stackexchange](https://math.stackexchange.com/questions/1143589/exact-sequence-of-groups-to-exact-sequence-sheaves) post. While not research level, after a week of no response, I figured I would ask it here.
For a topological group $G$ and a topological space $X$, denote... | https://mathoverflow.net/users/16639 | Exact sequence of groups to exact sequence of sheaves | The following is perhaps more of an extended comment than an answer. The sequence of sheaves is exact iff the quotient map $G\to H$ has a section over a neighborhood of every point (in fact, because of the group structure, it suffices to have a section over any single nonempty open set). In particular, for instance, th... | 7 | https://mathoverflow.net/users/75 | 197913 | 95,797 |
https://mathoverflow.net/questions/197918 | 16 | David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am interested in, $n=-2$.
What is the reason for this? Maybe I am missing something. Any references to results involving primes ... | https://mathoverflow.net/users/63939 | What is known about primes of the form $x^2-2y^2$? | Take any square free $1 \neq n \in \mathbb{N}$ and recall that $R\_n = \mathbb{Z}[\sqrt{n}]$ has a multiplicative norm function $N \colon R \to \mathbb{Z}$ given by $N(x + y\sqrt{n}) = x^2 -ny^2$ so a prime $p$ is of the required form iff $p$ is a norm in $R\_n$ (i.e $p$ is in the image of $N$).
Now take $n$ which is... | 19 | https://mathoverflow.net/users/38889 | 197921 | 95,800 |
https://mathoverflow.net/questions/107203 | 3 | The following is a theorem of Baumgartner, Hajnal, and Mate:
Suppose $J$ is a normal ideal on $\omega\_1$ which is nowhere $\omega\_1$-dense. Then for any sequence $\langle A\_\alpha : \alpha < \omega\_1 \rangle$ of $J$-positive sets, there is a sequence $\langle B\_\alpha : \alpha < \omega\_1 \rangle$ of pairwise di... | https://mathoverflow.net/users/11145 | Disjoint refinements in $P(\kappa)/J$ | In my [thesis](http://www.math.uci.edu/~meskew/thesis.pdf) I proved that my conjecture is false. It's not so simple-- the proof ties together chapter 2 and sections 5.1, 5.2, and 6.3.
| 4 | https://mathoverflow.net/users/11145 | 197924 | 95,802 |
https://mathoverflow.net/questions/197906 | 20 | One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between them. Call it $\mathcal{Cob}\_d$. It is well known that the connected components of the geometric realization of this ca... | https://mathoverflow.net/users/58952 | Super-cobordisms | There are a number of technical issues with making what you describe precise, for example: what precisely is a supermanifold with boundary? how can you glue/compose bordisms? etc. I am going to ignore these technicalities because I don't think it makes much of a difference to your question. Many of these technical issu... | 16 | https://mathoverflow.net/users/184 | 197932 | 95,804 |
https://mathoverflow.net/questions/172930 | 2 | Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this so? And if yes, what filtration should one choose?
The definition of semi-free I am using is: $X$ is semi-free if $X$ is... | https://mathoverflow.net/users/38075 | Does semi-free behave well under totalization | If $M\_\bullet$ is not bounded above, then the totalization may not be semi-free.
For example, let $k$ be a field, and $A=k[x]/(x^2)$, considered as a DG algebra concentrated in degree zero, with trivial differential. Then the chain complex
$$\dots\to A\to A\to A\to A\to\dots,$$
where all the differentials are multip... | 3 | https://mathoverflow.net/users/22989 | 197942 | 95,808 |
https://mathoverflow.net/questions/196082 | 14 | Given two unit-diameter disks tangent to a given line and to each other, determining a region bounded by two circular arcs and a line segment, is the Ford disk packing of that region the unique packing that covers as much total area as possible, among all ways of packing the region with disks tangent to the line?
As ... | https://mathoverflow.net/users/3621 | Is the Ford disk packing optimal? | There is an affirmative answer to a related question in which we view these disks as horodisks in the upper half-plane model of the hyperbolic plane. In this setting, it is natural to extend the Ford disk packing by applying (integer) horizontal translation and to add one more horodisk to the packing, namely, the shift... | 2 | https://mathoverflow.net/users/3621 | 197943 | 95,809 |
https://mathoverflow.net/questions/197935 | 2 | More precisely, The setting could be formulated as,
$min. F\_{\lambda}(p)$ over permutation matrices $P$
Here $F\_{\lambda}(p)$=$\lambda \*F\_{0}(p)+(1-\lambda)F\_{1}(p)$
where both $F\_{0}(p)$ and $F\_{1}(p)$ are **quadratic** (*Frobineous norm*) and $F\_{0}(p)$ is convex, $F\_{1}(p)$ is concave. Thus their com... | https://mathoverflow.net/users/68280 | Is there any algorithm can find local minima of nonconvex objective function in guaranteed polynomial time? | In general, without additional assumptions this will not be possible (barring P=NP).
In particular, from the (slightly edited) abstract of: K. G. Murty, S. K. Kabadi. [**Some NP-Complete problems in quadratic and nonlinear programming**, *Mathematical Programming*, 39(1987), 117-129](http://link.springer.com/content/... | 2 | https://mathoverflow.net/users/8430 | 197946 | 95,810 |
https://mathoverflow.net/questions/197945 | 4 | Consider a matrix $A\in{\bf M}\_{n\times m}({\mathbb R})$, whose entries are non-negative. Let $r$ be the rank of $A$.
It is well-known that $A$ decomposes as $x\_1y\_1^T+\cdots+x\_ry\_r^T$ with $x\_j\in{\mathbb R}^n$ and $y\_i\in{\mathbb R}^m$. But is this still true if we require in addition that $x\_1,\ldots,y\_r$... | https://mathoverflow.net/users/8799 | Non-negative decomposition of a non-negative matrix | This seems to be the well-known (and difficult) task of computing [the nonnegative rank of a matrix.](http://en.wikipedia.org/wiki/Nonnegative_rank_%28linear_algebra%29)
It seems that the true complexity of computing the nn-rank is unknown, while verifying whether nn-rank equals actual rank is NP-Hard (the Wikipedia ... | 6 | https://mathoverflow.net/users/8430 | 197947 | 95,811 |
https://mathoverflow.net/questions/196504 | 6 | Let $\mu\_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu\_n$ are measurable maps from a probability space $(\Omega,\mathcal{T},\mathbf{P})$ to the space of $M\_1(\mathbb R)$ equipped with the Borel-$\sigma$-algebra ge... | https://mathoverflow.net/users/43528 | Weak convergence of random measures | I found a way to prove the implication I was mostly interested in (the last statement implies the second): Below my argument, for the sake of completeness.
Fix a function $f$. A sequence of random variables $X\_n$ converges to a random variable $X$ in probability if and only if any subsequence $X\_{n\_m}$ has a furth... | 0 | https://mathoverflow.net/users/43528 | 197964 | 95,814 |
https://mathoverflow.net/questions/197917 | 24 | Numerical evidence shows the validity of the following identity
$$\int\limits\_0^z\frac{xdx}{\sin{x}\sqrt{\sin^2{z}-\sin^2{x}}}=\frac{\pi}{4\sin{z}}\ln{\frac{1+\sin{z}}{1-\sin{z}}},\tag{1}$$
if $0< z< \pi/2$.
How can it be proved? An indirect proof can be found in the paper <http://link.springer.com/article/10.1134%2FS... | https://mathoverflow.net/users/32389 | Interesting integral | Actually, I now think that the easiest method is to do this: Write $k=\sin z$, so that $|k|<1$, and make the substitution $x = \arcsin(k\sin\theta)$, where $0\le \theta\le \frac\pi2$. The integral becomes
$$
\int\_0^{\pi/2} \frac{\arcsin(k\sin\theta)}{k\sin\theta \,\,(1-k^2\sin^2\theta)^{(1/2)}}\ \mathrm{d}\theta
= \i... | 35 | https://mathoverflow.net/users/13972 | 197966 | 95,815 |
https://mathoverflow.net/questions/197948 | 1 | consider a domain $U\subset \mathbb{R}^n$ which is not bounded and denote its boundary by $\partial U$. (The boundary I'm confronted with is actually not very irregular. Think of a smooth manifold with corners.). Denote the heat kernel for the domain $U$ by $K\_{U}(t,x,y)$ and the heat kernel for $\mathbb{R}^n$ by $K(t... | https://mathoverflow.net/users/68283 | Heat kernel for non bounded domains | If such $H$ exists, the $H(t,x,y)\le K(t,x,y)$ due to the maximum principle.
| 0 | https://mathoverflow.net/users/14551 | 197974 | 95,817 |
https://mathoverflow.net/questions/195675 | 8 | Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see <http://en.wikipedia.org/wiki/Mountain_pass_theorem> for the precise statement of the theorem). I wonder if there is a generalization of t... | https://mathoverflow.net/users/42326 | Mountain Pass theorem for minimization problems with constraints | You can generalize the mountain pass theorem if the constraints consist of a Banach manifold.
This is because you can construct a pseudo gradient flow on a Banach manifold, which is required in the proof of the theorem.
Check Chapter 27 of 'Nonlinear Functional Analysis' by K. Deimling.
For example, in the paper 'Sta... | 5 | https://mathoverflow.net/users/68306 | 197981 | 95,820 |
https://mathoverflow.net/questions/197937 | 3 | For $n>2$, are there norms $\parallel.\parallel\_{a}$ and $\parallel.\parallel\_{b}$ on $M\_{n}(\mathbb{R})$ with the following property:
>
> $A\in M\_{n}(\mathbb{R})$ is singular if and only if $\parallel A \parallel\_{a}=\parallel A \parallel\_{b}$
>
>
>
For $n=2,\;$ these norms are $\parallel A \parallel\_... | https://mathoverflow.net/users/36688 | A norm description for singular matrices | (This answer expands the ideas in the comments by Mikael de la Salle and myself).
There is a family of matrix norms that can be defined as follows: given $p \geq 1,k \leq n$,
$$
\|A\|\_{p,k} = \left(\sum\_{i=1}^k \sigma\_i(A)^p\right)^{1/p},
$$
where we denote by $\sigma\_i(A)$ the $i$th [singular value](https://en.w... | 7 | https://mathoverflow.net/users/1898 | 197982 | 95,821 |
https://mathoverflow.net/questions/197983 | 7 | I am looking for survey-books on open math (esp. probability) problems from engineering fields but phrased in mathematical language.
There are hundreds of specialized math-engineering books out there, but I haven't found an introductory survey of math-engineering problems.
Given the diversity of problems in engine... | https://mathoverflow.net/users/40793 | Survey of Engineering Problems for Mathematicians | There is a (perhaps somewhat dated) nice little book devoted to open problems in communications (coding and information theory), system and control theory, and computer science where open problems are motivated. Some of these problems were actually solved by the participants at the 3 workshops in the 1980s, on which th... | 4 | https://mathoverflow.net/users/17773 | 197994 | 95,827 |
https://mathoverflow.net/questions/196539 | 10 | The circle in homotopy type theory $\mathbb{S}^1$ is a higher inductive type freely generated by the following constructors:
$\mathsf{b} : \mathbb{S}^1$ and $\mathsf{loop} : \mathsf{b} = \mathsf{b}$.
The sphere $\mathbb{S}^2$ is freely generated by the following constructors $\mathsf{b'} : \mathbb{S}^2$ and $\mathsf{... | https://mathoverflow.net/users/61413 | How to proceed with a type-theoretic proof that $\Sigma \mathbb{S}^1 \simeq \mathbb{S}^2$? | I'm not sure if it is the most elegant way, but it is certainly the most direct. So, we need to define the image of $\mathrm {surf}$ as an element of $refl\_{\mathrm N} = refl\_{\mathrm N}$. We proceed as follows. Let $p \equiv m(\mathrm b) : \mathrm N = \mathrm S$. Consider the inverse path $p^{-1}$. We have a proof (... | 4 | https://mathoverflow.net/users/10605 | 197995 | 95,828 |
https://mathoverflow.net/questions/197959 | 22 | Let $C$ : ${\mathbb N}\longrightarrow {\mathbb N}$ be Collatz's map defined by $C(n) = 3n+1$ if $n$ is odd, and $C(n)=n/2$ if $n$ is even. Then according to Collatz's conjecture, we should have $C^k (n)=1$, for all $n>0$ and $k$ large enough.
Assume that the conjecture holds and for $n>0$ define the top of the orbit ... | https://mathoverflow.net/users/4767 | A question on Collatz's conjecture:proportion of "low flying" orbits | If the sequence $(T(N))\_{N \in \mathbb{N}}$ converges and the limit is
not equal to $0$, this would imply either positive predecessor density for $1$, cf. e.g.
Günther J. Wirsching, [*On the problem of positive predecessor density in $3n+1$ dynamics*](https://www.aimsciences.org/journals/displayArticles.jsp?paperID=... | 13 | https://mathoverflow.net/users/28104 | 197997 | 95,829 |
https://mathoverflow.net/questions/197898 | 3 | I posted this question in <https://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close> a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F\_q$ with minimum distance $d$ and number of minimum weight codewords $N\_d$.
How many ways can you select codewords $c\_1,\dots,c\_T$ (assume $T\ll q... | https://mathoverflow.net/users/10035 | Picking codewords that are close | Edited:
@Turbo: You're right. I was assuming that the two codewords must be nonzero. In general, when one codeword is zero, one gets a weaker lower bound of the form
$N\_d {q^k \choose T}$
and the rest of the argument follows with a linear $N\_d$ instead of a quadratic factor $N\_d^2/2.$
A special case (original an... | 1 | https://mathoverflow.net/users/17773 | 198002 | 95,832 |
https://mathoverflow.net/questions/197996 | 4 | Let $f:S^1\to S^1$ be an orientation-preserving circle diffeomorphism with irrational rotation number (see [here](http://en.wikipedia.org/wiki/Rotation_number)). Then the system $(S^1,f)$ admits a unique invariant measure, say $\mu\_f$.
Let $\displaystyle \lambda(f,x)=\lim\_{n\to\infty}\frac{1}{n}\log D\_xf^n$ be th... | https://mathoverflow.net/users/11028 | Lyapunov exponent for circle diffeomorphisms | Note that $\int \log D\_xf^n \, dm(x)\le \log \int D\_xf^n \, dm(x)=\log 1=0$ for all $n\ge 1$. Therefore (by uniform convergence for unique ergodic system)
$\displaystyle \lambda(f)=\int \lambda(f,x) dm(x)=\int\lim\_{n\to\infty}\frac{1}{n}\log D\_x f^n dm(x)=\lim\_{n\to\infty}\frac{1}{n}\int\log D\_x f^n dm(x)\le 0$... | 3 | https://mathoverflow.net/users/11028 | 198004 | 95,834 |
https://mathoverflow.net/questions/198001 | 1 | Let $R$ be a commutative ring and $X$ a topological space. Define a *sheafy cohomology theory* (see [here](http://www-math.sp2mi.univ-poitiers.fr/~sarti/corso_Perego.pdf)) to be a collection of functors $\mathrm{H}^q:\mathrm{Sh}(X;R\mathrm{Mod})\to R\mathrm{Mod}$ such that the following conditions are satisfied:
* If... | https://mathoverflow.net/users/nan | Axioms for sheaf cohomology | With respect to the updated question: yes, there are other cohomology theories. For example, if $H$ is ordinary sheaf cohomology, then we can define a new sheaf cohomology theory $K$ by $K^q({\cal F}) = H^q({\cal F}) \times H^{q-1}({\cal F})$. (This is a special instance of a hypercohomology construction which is genui... | 5 | https://mathoverflow.net/users/360 | 198005 | 95,835 |
https://mathoverflow.net/questions/197993 | 7 | From the [Wikipedia article on Primitive recursive arithmetic](https://en.wikipedia.org/wiki/Primitive_recursive_arithmetic):
>
> *"Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers. It was first proposed by Skolem[1] as a formalization of his finitist conception of t... | https://mathoverflow.net/users/10110 | Primitive recursive arithmetic via universal algebra | According to unpublished notes by Gavin Wraith ("Notes on arithmetic universes and Gödel incompleteness theorems" (1985)), PRA can be described as an equational theory or as a Lawvere theory, and is abstractly characterized as initial among all Lawvere theories whose generating object is a parametrized natural numbers ... | 10 | https://mathoverflow.net/users/2926 | 198010 | 95,838 |
https://mathoverflow.net/questions/198025 | 0 | [Ore's theorem](http://en.wikipedia.org/wiki/Ore%27s_theorem) states that in a finite graph $G$ with $|V(G)|=n$, there is a Hamiltonian path, provided that the sums of the degrees of 2 distinct, non-adjacent vertices is $\geq n$.
For countable graphs, such a statement cannot hold: consider the disjoint union of two c... | https://mathoverflow.net/users/8628 | Ore's theorem for countable graphs | And then here is an obvious counterexample: take a star-like tree with one central vertex of countable degree (and, if you still insist on two non-adjacent vertices, make each ray of length $2$).
| 2 | https://mathoverflow.net/users/44953 | 198026 | 95,844 |
https://mathoverflow.net/questions/198017 | 1 | Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a graded version for this? I mean:
Let $R$ be a graded ring. Let $M$ be a graded $R$-module.
>
> If $Ext\_R^1(R/I,E... | https://mathoverflow.net/users/47763 | Graded version of Baer's Criterion | To get the correct graded version you have to observe all possible shifts. More precisely:
>
> Let $G$ be an abelian group, let $R$ be a $G$-graded ring, and let $M$ be a $G$-graded $R$-module. Then, the following statements are equivalent:
>
>
> *(i)* $M$ is injective;
>
>
> *(ii)* For every $g\in G$, every mo... | 4 | https://mathoverflow.net/users/11025 | 198031 | 95,845 |
https://mathoverflow.net/questions/197949 | 6 | I try to prove $L\_{SO}=\mathrm{HOD}$, where $L\_{SO}$ is second-order constructible universe which has similar definition with $L$ but it uses second-order definability rather than the first-order definability, and I found [the answer in MO](https://mathoverflow.net/a/156949/48041). Also, I found the referred article ... | https://mathoverflow.net/users/48041 | Axiom of choice and the equality between second-order constructible universe and HOD | The equality $L\_{SO}=HOD$ can not be proved just in $ZF$. This is proved in the paper ``The consistency of the theory $ZF+L^1\neq HOD$'' by Szczepaniak.
Here $L^1$ refers to what you named $L\_{SO}$. The idea of the proof is as follows:
$(1)$ If two models of $ZF$ have the same sets of ordinals, then they have the... | 8 | https://mathoverflow.net/users/11115 | 198033 | 95,846 |
https://mathoverflow.net/questions/198019 | 6 | This question came up while going through the application of Eisenstein criterion: The $p$-th cyclotomic polynomial after changing the variable $x$ to $(x+1)$ satisfies Eisenstein criterion. That is the minimal polynomial of $\zeta\_p-1$ is an Eisenstein polynomial.
Now let us take a general algebraic number field *... | https://mathoverflow.net/users/22878 | Do all algebraic number fields arise from Eisenstein polynomials? | As already indicated by the comment of Mostafa, the criterion is that $K$ is totally ramified at some prime $p$. Mostafa's comment shows that this is necessary. To see that it is also sufficient,
take any integral element $\alpha$ of $K$
whose $p$-adic valuation is $1/n$ (if the valuation is normalizied to be 1 at $p$)... | 13 | https://mathoverflow.net/users/21146 | 198034 | 95,847 |
https://mathoverflow.net/questions/198018 | 2 | Let $Psh(\mathcal{C})$ be the category of simplicial presheaves equipped with the projective model structure. The cartesian product between two representables presheaves is clearly again representable and hence cofibrant, what about a more general statament?
>
> 1) Is it true that the the cartesian product between ... | https://mathoverflow.net/users/41970 | Cartesian products between cofibrant simplicial presheaves | If $\mathcal{C}$ has finite products, then the class of projective cofibrations is also closed under finite products. Indeed, since the cartesian product in the category of simplicial presheaves preserves colimits in each variable, it suffices to check the claim on the generating projective cofibrations, i.e. that for ... | 2 | https://mathoverflow.net/users/11640 | 198035 | 95,848 |
https://mathoverflow.net/questions/198039 | 5 | Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V \rightarrow M$ be the projectivization of $V$.
$\textbf{Question}:$ Is there a formula
for $c(T\mathbb{P}V)$, the t... | https://mathoverflow.net/users/4463 | Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle? | No, your formula is not correct. You have to take into account the Chern classes of $V$. The relative tangent bundle $T\_{\mathbb{P}V/M}$ is given by the so-called Euler exact sequence
$$0\rightarrow \mathscr{O}\_{\mathbb{P}V}\rightarrow \pi ^\*V\otimes \gamma^\* \rightarrow T\_{\mathbb{P}V/M}\rightarrow 0\ ,$$
while $... | 8 | https://mathoverflow.net/users/40297 | 198042 | 95,850 |
https://mathoverflow.net/questions/198049 | 11 | Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let $s \in S$ be a point. Let $\beta \colon X \to S$ be the blowup of $s \in S$. Suppose that $H^{i}(S, T\_{S})$ is known f... | https://mathoverflow.net/users/21815 | Deformations of a blowup | The answer is the following and can be found in Hartshorne's book *Deformation Theory*, see in particular Exercise 10.5 page 83.
We work over an algebraically closed field $k$. Then there is an exact sequence of sheaves $$0 \to \beta\_\*T\_X \to T\_S \to k\_s \oplus k\_s \to 0,$$ inducing an exact sequence in cohomol... | 11 | https://mathoverflow.net/users/7460 | 198050 | 95,853 |
https://mathoverflow.net/questions/198032 | 4 | If $G = (V,E)$ is a graph, then a $\omega$-*path* is an injective map $p:\omega\to V$ such that $\{p(k),p(k+1)\}\in E$ for all $k\in \omega$. In a similar fashion, we define a $\mathbb{Z}$-*path*.
Is there a graph $G$ with $V(G) = \omega$ and the following properties?
1. There is no surjective $\mathbb{Z}$- or $\om... | https://mathoverflow.net/users/8628 | Countable hypo-hamiltonian graph | Yes, Figure 2 of Carsten Thomassen's paper "Planar and infinite hypohamiltonian and hypotraceable graphs" ([doi:10.1016/0012-365X(76)90071-6](http://dx.doi.org/10.1016/0012-365X(76)90071-6)) presents a graph that does not have a 2-way infinite hamiltonian path, but such that every vertex-deleted subgraph does have such... | 10 | https://mathoverflow.net/users/68305 | 198068 | 95,858 |
https://mathoverflow.net/questions/198067 | 7 | I'm trying to understand the result given in the first box at slide 45 of [this talk](http://homepages.math.uic.edu/~jbaldwin/cms07.pdf). Specifically:
1) What is the source cited? I have not been able to find any article by Keisler, Chudnovsky and/or Shelah corresponding to the situation.
2) Is this an alternate p... | https://mathoverflow.net/users/35734 | Alternate proof of Morley's theorem? | Regarding (2), some evidence that Baldwin refers to some sort of $L\_{\omega\_1 \omega}$ version of Morley's theorem, rather than just an alternate proof making use of $L\_{\omega\_1 \omega}$ machinery, comes from a [1970 survey by Keisler himself](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0141.0150.ocr.pdf... | 5 | https://mathoverflow.net/users/4137 | 198075 | 95,860 |
https://mathoverflow.net/questions/197901 | 5 | Let $(\alpha\_k)$ be a sequence of positive numbers and let $(Y\_k)$ be a sequence of independent random variables $Y\_k \sim \text{Gamma}(\alpha\_k,1)$. Set $X\_n=\dfrac{Y\_n}{\sum\_{i=1}^nY\_i}$.
**(edit)** The title is not appropriate: $(X\_1, ..., X\_n)$ is not a Dirichlet vector, because $X\_k=\dfrac{Y\_k}{\sum\... | https://mathoverflow.net/users/21339 | Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, ..., X_n)$ | The answer to the second question is no too.
Set $D\_k^n=\dfrac{Y\_k}{\sum\_{i=1}^nY\_i}$. The *neutrality property* of Dirichlet vectors says that $D\_n^n=:X\_n$ is independent of $(D\_1^{n-1}, \ldots, D\_{n-1}^{n-1})$. But $X\_k=\dfrac{D\_k^{n-1}}{D\_1^{n-1}+\ldots+D\_k^{n-1}}$ for every $k=1, \ldots, n-1$, theref... | 2 | https://mathoverflow.net/users/21339 | 198077 | 95,861 |
https://mathoverflow.net/questions/198093 | 4 | If two topological spaces are weak homotopy equivalent to each other, are their Cech cohomology groups the same?
| https://mathoverflow.net/users/68356 | Weak homotopy equivalence and Cech cohomology | $$T = \left\{ \left( x, \sin \frac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,y)\mid y\in[-1,1]\}$$
This has trivial homotopy groups in degrees $\ge1$ but according to Wikipedia nontrivial Čech cohomology in degree 1.
| 4 | https://mathoverflow.net/users/39082 | 198099 | 95,865 |
https://mathoverflow.net/questions/198094 | 0 | I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement:
Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ and $b \in (0,1)$. Then there exists some $x\_0\in\mathbb R$ such that for all $x \geq x\_0$:
$$\left(1-\frac{1}{b \ln^{... | https://mathoverflow.net/users/32355 | Proving a complicated inequality with powers of logarithms | When $x\to \infty$ we have
$$\left(1-\frac1{b\ln^{1+\epsilon}x}\right)^p = 1 -\frac p{b\ln^{1+\epsilon}x} + o\left(\frac1{\ln^{1+\epsilon}x}\right),$$
and
$$\frac1{\ln^{\epsilon/2}\left(bx + \frac x{\ln^{1+\epsilon}x}\right)}= (\ln bx)^{-\epsilon/2} \left(\frac1{1+ \frac1{\ln bx} \ln\left(1 + \frac1{b\ln^{1+\epsilon}x... | 3 | https://mathoverflow.net/users/59023 | 198104 | 95,868 |
https://mathoverflow.net/questions/198098 | 40 | Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be triangularized and hence be described by a finite amount of combinatorial data.
In higher dimensions this argument doesn't work... | https://mathoverflow.net/users/37059 | Are there only countably many compact topological manifolds? | It was shown in
J. Cheeger and J. M. Kister, *Counting topological manifolds*. Topology 9, 1970 149–151.
that there are only countably many compact manifolds up to homeomorphism (even allowing boundaries).
Here is a [link](http://www.sciencedirect.com/science/article/pii/0040938370900364) to the article.
| 41 | https://mathoverflow.net/users/8176 | 198105 | 95,869 |
https://mathoverflow.net/questions/191465 | 9 | ***Reformulation of the question (see below for the original question):*** Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field extensions $L\supseteq K$ such that $L\otimes\_K D$ is a division algebra?
I have encountered this question in the cont... | https://mathoverflow.net/users/35394 | Division algebras over extension fields / reducibility of $G$-modules | Let $K$ be a number field and $D$ a finite dimensional central division algebra over $K$. I believe that one can characterize the field extensions $L$ of $K$ for which $D\otimes\_K L$ is a division algebra. I am not sure how useful this particular characterization will be in the context of the representation theory of ... | 4 | https://mathoverflow.net/users/nan | 198110 | 95,872 |
https://mathoverflow.net/questions/198100 | 1 | There seems to be a lot of theorems allowing to prove restricted cases of this (eg. uniformization, classification theorem for compact surfaces) . Intuitively, it seems true, but I've never seen a proof of the general case.
| https://mathoverflow.net/users/27712 | Do all surfaces (2d riemanian manifolds) admit constant curvature? | This is a standard consequence of the uniformization, but I agree that locating a reference may be a challenge. You can find details in my survey <http://arxiv.org/abs/1306.1256>, see Theorem 2.2 where it is shown that any open (smooth connected) 2-manifold admits a complete metric of constant negative curvature. The s... | 4 | https://mathoverflow.net/users/1573 | 198112 | 95,873 |
https://mathoverflow.net/questions/198111 | 3 | Let $l$ be a positive integer. Does the matrix
$$
M\_l \ := \ \left( \binom{l-(2p+1)}{j} \right)\_{0\leq p,j \leq[(l-1)/2]}
$$
have nonzero determinant?
| https://mathoverflow.net/users/68368 | Invertibility of a matrix whose entries are certain binomial coefficients | There is a nice result of Gessel and Viennot that computes your determinant in terms of NE lattice paths. The original paper is available [here](http://www.sciencedirect.com/science/article/pii/0001870885901215). Aigner and Ziegler also give a nice exposition of this result in their [Proofs From the Book](http://rads.s... | 6 | https://mathoverflow.net/users/66536 | 198113 | 95,874 |
https://mathoverflow.net/questions/197969 | 6 | Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (**Edit 1:** "split" has been removed from "maximal torus"; the building of $U\_3(L/F)$, where $L/F$ is a ramified quadratic extensio... | https://mathoverflow.net/users/30726 | When are toral orbits in buildings the difference of fixed-sets? | (Editted: a "weaker" example about $GL\_3$ at the end)
If I didn't make a mistake in my computation, then the second question doesn't hold for $G=Sp\_4$, as it doesn't hold for any $\mathbb{Q}\_p$. Allow me to use $F$ as my p-adic field and $k$ its residue field. Let $V/\_F$ be spanned by $e\_2,e\_1,f\_1,f\_2$ with the... | 5 | https://mathoverflow.net/users/31327 | 198117 | 95,876 |
https://mathoverflow.net/questions/198097 | 4 | I want to show:
Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align}
H^1\_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)
\end{align}
is compact.
I was able to show that \begin{align}|u(r)|\leq C R^{\frac{-(N-1)}{2}} \|\nabla u\|\_2^{\frac{1}{2}} \|u\|\_2^{\frac{1}{2}}\leq \hat C R^{\frac{-(... | https://mathoverflow.net/users/47482 | Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$ | It's Strauss embedding theorem for radially symmetric functions, proven here:
W. A. Strauss, *Existence of solitary waves in higher dimensions*, Commun. Math.
Phys. 55 (1977), 149-162.
| 3 | https://mathoverflow.net/users/6101 | 198123 | 95,878 |
https://mathoverflow.net/questions/198118 | 4 | Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical
module, $K$. By [Proposition 3.3.18](https://books.google.com/books?id=LF6CbQk9uScC&pg=PA116&dq=Proposition+3.3.18+Let+R+be+a+Cohen%0CMacaulay+ring+and&hl=en&sa=X&ei=6dDoVPmfGsblaK2tgYAG&ved=0CB8Q6AEwAA#v=onepage&q=Propositio... | https://mathoverflow.net/users/47763 | canonical module can be identified with an ideal. how can one reach that ideal? | Well, there is a non-canonical way.
*try random embeddings until it works.*
Let me explain.
Strategy
--------
Say we are given a module $M$ with generators $x\_1, \ldots, x\_n$. We know that the module is an ideal (by some general nonsense) but we don't know how to embed it as an ideal of $R$. Thus we need to ... | 6 | https://mathoverflow.net/users/3521 | 198129 | 95,881 |
https://mathoverflow.net/questions/198124 | 2 | Is there a way to deduce Zorn's lemma from Zermelo theorem (that any set may be well ordered), which is essentially shorter then deduction of Zorn's lemma from the usual form of Axiom of Choice?
| https://mathoverflow.net/users/4312 | Zorn's lemma via Zermelo theorem | *I assume the "usual proof" goes by defining the obvious chain of elements $p\_\eta$ by induction on $\eta$, and then arguing that that constitutes a chain with no upper bound.*
I'm not sure if this counts, but: let $<$ be a well-ordering of the set of chains through $\mathbb{P}$. Now we can define a sequence of chai... | 1 | https://mathoverflow.net/users/8133 | 198130 | 95,882 |
https://mathoverflow.net/questions/196610 | 6 | I am trying to make sense of what is written in Rezk's draft <http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf>
In particular, I am referring to Proposition 2.3, which is there stated without proof.
According to different models for the homotopy colimit functor on $\text{sSet}^J$, we may or may not f... | https://mathoverflow.net/users/57280 | Descent properties of spaces | For the first problem, as I wrote in the comments, the author is implicitly using the language of $(\infty,1)$-categories, where it does make sense to speak of such a functor.
To understand why, you can consult any of the many introductions to $(\infty,1)$-category theory.
The rest of this answer deals with the secon... | 3 | https://mathoverflow.net/users/2503 | 198135 | 95,885 |
https://mathoverflow.net/questions/198136 | 5 | The asymptotic number of
[square-free numbers](http://en.wikipedia.org/wiki/Square-free_integer#Distribution)
$\le n$ is $Q(n) = 6n/\pi^2 + O(\sqrt{n})$.
Because
[$\zeta(2)=\pi^2/6$](http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html),
$Q(n) \approx n/\zeta(2)$.
[OEIS A004709](http://oeis.org/A004709)
says th... | https://mathoverflow.net/users/6094 | Square-free grows as $6n/\pi^2$: $k$-th free? | Yes, it works in much the same way for any $k$. Here's an elementary proof.
Let $Q\_k(n)$ be the number of $k$-th power free integers $\leq n$.
Then
$$
Q\_k(n) = \sum\_{d^k \leq n} \mu(d) \lfloor n/d^k \rfloor
= \sum\_{d^k \leq n} \mu(d) \, (n/d^k + \theta\_d)
$$
for some $\theta\_d \in [0,1)$. Hence
$$
\Bigl| \, Q\... | 9 | https://mathoverflow.net/users/14830 | 198139 | 95,888 |
https://mathoverflow.net/questions/198085 | 5 | The stirling number of the second kind $S(n,k)$ counts the number of partitions of the set $[n]$ into $k$ non-empty parts. I found a definition for the numbers called the $r$-associated stirling numbers of the second kind in wikipedia. These count the number of partitions of $[n]$ into $k$ non-empty parts such that all... | https://mathoverflow.net/users/24478 | Stirling numbers of the second kind with maximum part size | This is a routine application of the exponential formula. If $S$ is any subset of the positive integers and $f\_S(n)$ is the number of partitions of $[n]$ into parts all belonging to $S$, then
$$ \sum\_{n\geq 0} f\_S(n) \frac{x^n}{n!}= \exp \sum\_{i\in S}\frac{x^i}{i!}. $$
Thus in your case we get $\exp \sum\_{i=1}^r ... | 5 | https://mathoverflow.net/users/2807 | 198142 | 95,890 |
https://mathoverflow.net/questions/197872 | 1 | I'm interested in the first basic case of excess intersection in intersection theory:
Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap T$ contains an effective 1-cycle $D$ as its 1-dimensional part. In other words, $Z$ defines a Cartier divisor on $S$.
... | https://mathoverflow.net/users/56505 | Non-proper intersection of surfaces | The answer is NO. Take $X=\mathbb{P}^4$, $S,T$ two smooth quadrics. Then $(S\cdot T)=4$, the normal bundles are $N\_S=\mathcal{O}\_S(1)\oplus \mathcal{O}\_S(2)$, and same for $T$. If $S$ and $T$ are general, $Z=0$. If they are given by
$$S:\ X=0\ ,\ YU+TV=0\quad;\quad T:\ Y=0\ ,\ XV+TU=0$$(coordinates $(X,Y,T,U,V)$ on... | 4 | https://mathoverflow.net/users/40297 | 198159 | 95,899 |
https://mathoverflow.net/questions/198128 | 1 | let $u(t,x)$ be a bounded smooth solution of the heat equation $u\_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: u(t,x)>0\}$. Suppose that $(\{t\_1\} \times R^2) \cap V$ has only one open connected component which is also bounded. ... | https://mathoverflow.net/users/42326 | Strong maximum principle for the heat equation in non-cylindrical domains | This is certainly possible, and it seems rather obvious on physical grounds. Consider an initial condition where you have two hot regions connected by a thin corridor which is also hot, and the surrounding space is cold. It is clear that the connecting corridor will cool rapidly.
| 1 | https://mathoverflow.net/users/12120 | 198167 | 95,902 |
https://mathoverflow.net/questions/198127 | 2 | If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave similar to the ones for $q=0$ which are given by $l(l+1)$. Unfortunately, I only found asymptotic estimates for the free case an... | https://mathoverflow.net/users/68375 | Asymptotic behaviour of eigenvalues | I would say that the eigenvalues of $-\Delta+q$ stay within $\|q\|\_{\infty}$ of eigenvalues of $-\Delta$ (and, as Noam Elkies says, note the multiplicity: the main term of asymptotics will be $\lambda\_n\sim n$, as $l(l+1)\sim l^2$ is an eigenvalue of $-\Delta$ of multiplicity $2l+1$.)
If I'm not mistaken, you can r... | 0 | https://mathoverflow.net/users/31371 | 198170 | 95,903 |
https://mathoverflow.net/questions/198091 | 1 | I am wondering why the first well known example of non-tempered irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the non-tempered representation of $U(3)$ using the theta lift.
I know all irreducible representation of $U(1)$ should be tempered be... | https://mathoverflow.net/users/29422 | Is there a non-tempered representation of U(2)? | The admissible representations of U(2) over a p-adic field are fairly straightforward since they essentially come from those of SL(2) or a the norm one elements of the quaternions.
Let me add (two days later) that the point is not that the representation of U(3) that arises from the theta-lift is non-tempered but ra... | 1 | https://mathoverflow.net/users/7448 | 198173 | 95,904 |
https://mathoverflow.net/questions/198166 | 3 | Let $\{v\_1, \dotsc, v\_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R\_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows:
$$R\_m(M) := \sum\_{i,j=1}^m (v\_i^\*M v\_j) E\_{ij}$$
where $E\_{ij} := e\_i e\_j^\*$ is the all-zeroes matrix with entry 1 at position $(i,j)$. In other wor... | https://mathoverflow.net/users/37211 | Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary? | When the (general) question is rephrased in less basis-dependent language, I believe that it translates to this: Let $\mathrm{U}(d)$ act on $V = \mathbb{C}^d$ in the usual way, and consider the $n$-fold product $G = \mathrm{U}(d)\times \mathrm{U}(d)\times\cdots\times \mathrm{U}(d)$ acting by the usual tensor product on... | 8 | https://mathoverflow.net/users/13972 | 198175 | 95,905 |
https://mathoverflow.net/questions/198144 | 10 | Every (not-necessarily invertible) map $f$ from $[n]:=\{1,2,,,,.n\}$ to itself determines a linear map $L\_f$ from ${\bf R}^n$ to itself that sends the basis vector $e\_k$ to $e\_{f(k)}$ for $1 \leq k \leq n$.
If $f$ and $g$ are two self-maps of $[n]$ for which the associated linear maps $L\_f$ and $L\_g$ are conjug... | https://mathoverflow.net/users/3621 | Distinguishing combinatorial maps by their linearizations | As Benjamin Steinberg comments, the problem reduces to the following:
represent $f$ by the digraph with vertices $[n]$ and edges $i\to
f(i)$. Delete all the cycles. What remains is an acyclic digraph $D$
that defines a nilpotent matrix $N$ whose rows and columns are indexed
by the vertices of $D$. Namely, $N\_{uv}=1$ i... | 4 | https://mathoverflow.net/users/2807 | 198180 | 95,907 |
https://mathoverflow.net/questions/198182 | 1 | There is a well-known structure theorem for locally compact non discrete topological division algebras, see here
<https://math.stackexchange.com/q/1160086/187521>
(I repost it here because I think it is more suitable given the nature of the question) and the proof of this theorem generally always uses the existence... | https://mathoverflow.net/users/41142 | Structure of locally compact non discrete topological division algebras without the use of Haar measure | I am a little surprised that you think it's "intuitive" that Haar measure *must* be used.
In section 9.13 of Jacobson's "Basic Algebra II" this result is proved for totally disconnected division algebras. In section 27 in Chapter IV of Warner's "Topological Fields" Theorem 27.2 covers the connected case. Neither book... | 3 | https://mathoverflow.net/users/3272 | 198184 | 95,908 |
https://mathoverflow.net/questions/196523 | 5 | In $\S$ 9.3 of the book "Mirror symmetry" (Vafa, Zaslow eds.) the authors formulate the following general localization principle for computation of integrals with respect to both even and odd variables: Assume there is some supersymmetry transformation of variables. **Then the integral becomes localized on the field co... | https://mathoverflow.net/users/16183 | Localization principle in supersymmetry | I think what you are looking for is Theorem 1 in "Supersymmetry and Localization" by Schwarz and Zaboronsky.
| 2 | https://mathoverflow.net/users/11437 | 198189 | 95,909 |
https://mathoverflow.net/questions/198188 | 0 | I know that the points of an elliptic curve over $\mathbb{Q}$, $\mathbb{R}$ or other field $K$ form a group, particularly the most common example to explain the naive way is with this curve $y^2=x^3-x$, internally what is happening is that line with coefficients over the ground field intersects with $K$-rational points... | https://mathoverflow.net/users/91023 | Hyperelliptic curve of genus 2 over R | Basically, the problem is that there are really five intersection points with your line (two of them are complex non-real). If you could make a group in the same way as for elliptic curves, then this would work over arbitrary fields, including the complex numbers. But when you have three more points of intersection ins... | 7 | https://mathoverflow.net/users/21146 | 198191 | 95,910 |
https://mathoverflow.net/questions/198190 | 2 | The [Fermat-Catalan conjecture](http://en.wikipedia.org/wiki/Fermat%E2%80%93Catalan_conjecture) states that there are only finitely many sex-tuples $(a, b, c, d, e, f)$ of positive integers such that
(1) $a^d + b^e = c^f$,
(2) $\gcd(a, b, c) =1$,
(3) $\frac{1}{d} + \frac{1}{e} + \frac{1}{f} \lt 1$.
Here, I have... | https://mathoverflow.net/users/34490 | Examples that the Fermat-Catalan conjecture does not cover | There are three cases when $1/D+1/E+1/F=1$, which give $(D,E,F)=(2,3,6)$, $(4,4,2)$, and $(3,3,3)$. In these three cases the equations define curves that are in fact elliptic curves of rank $0$, and the only solutions are the obvious ones where one of $A$, $B$, $C$ is $0$.
The case $1/D+1/E+1/F>1$ is much more inter... | 7 | https://mathoverflow.net/users/4140 | 198193 | 95,912 |
https://mathoverflow.net/questions/198070 | 3 | Let $k$ be a finite extension of the p-adic number field $Q\_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as
$1\rightarrow G\_{aff}\cap G\_{ant}\rightarrow G\_{aff}\times G\_{ant}\rightarrow G \rightarrow 1,$ where $G\_{aff}$ is the sma... | https://mathoverflow.net/users/9401 | Exactness on rational points of algebraic groups | YES to Question 1. For an arbitrary homomorphism $\phi\colon G\to F$ of connected algebraic groups over $k$, not necessarily affine, where $k$ is a $p$-adic field or $k=\mathbb{R}$, the image $\phi(G(k))$ is closed in $F(k)$.
*Proof.* If $G$ is a connected $k$-group, and $X=G/H$ is a homogeneous space of $G$, then ev... | 2 | https://mathoverflow.net/users/4149 | 198194 | 95,913 |
https://mathoverflow.net/questions/157357 | 17 | One of the most basic examples in noncommutative geometry is the so-called *noncommutative torus*, denoted here by $ \mathbb{T}\_{\theta} $. As far as I know, there are several equivalent constructions of it:
1. as the $ C^{\*} $-algebra of a foliation;
2. as a crossed-product $ C^{\*} $-algebra;
3. as a universal $ ... | https://mathoverflow.net/users/24078 | Realisation of the noncommutative torus as a universal $ C^{*} $-algebra | According to what I have seen in the literature so far, the standard procedure consists of two main steps:
* Prove the existence of a universal $ C^{\*} $-algebra $ A\_{\theta} $ generated by two unitaries $ u $ and $ v $ that satisfy
$$
u v = e^{2 \pi i \theta} v u.
$$
**Note:** We are assuming that $ \theta $ is ir... | 13 | https://mathoverflow.net/users/50614 | 198199 | 95,914 |
https://mathoverflow.net/questions/198187 | 3 | Let $n \ge k \ge t \in \mathbb{N}$, and consider a universe $U$ of size $n$. Let $\mathcal{F}$ be a family of $k$-subsets of $U$, such that every $t$-subset of $U$ is contained in at least one member of $\mathcal{F}$. My question is: How small can $\mathcal{F}$ be? In other words, what is the smallest possible size of ... | https://mathoverflow.net/users/24226 | Minimal family of k-sets containing all t-sets | I think it is an optimal version of a covering with index $1$.
A $t$-$(n,k,\lambda)$ *covering* is an ordered pair $(U,\mathcal{B})$ of a finite set $U$ of cardinality $n$ and a finite set $\mathcal{B}$ of $k$-subsets of $U$ such that every $t$-subset appears as a subset in at least $\lambda$ elements of $\mathcal{B}... | 3 | https://mathoverflow.net/users/27829 | 198201 | 95,915 |
https://mathoverflow.net/questions/198213 | 7 | Is it true that
$$
||f\*g||\_p \le ||\,|f|^\* \* |g|^\*||\_p\quad ?
$$
where $|f|^\*$ and $|g|^\*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ and $g$ this is true?
| https://mathoverflow.net/users/68409 | Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement? | Yes, this follows from the [Riesz rearrangement inequality](http://en.wikipedia.org/wiki/Riesz_rearrangement_inequality):
$$
\int\_{\mathbb R^n} h(x)(f\*g)(x)\, dx \le \int\_{\mathbb R^n} h^\*(x) (f^\* \*g^\*)(x)\, dx
$$
(We can assume that all functions are $\ge 0$.) Since $\|h^\*\|\_q=\|h\|\_q$, this shows that
$$
\|... | 7 | https://mathoverflow.net/users/48839 | 198220 | 95,920 |
https://mathoverflow.net/questions/198022 | 2 | Consider the representation $L\_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L\_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L\_U M=U M U^{\dagger} $, $\forall M\in L(\mathbb{C}^n)$, $\forall U\in U(n)$.
I know that this representation is reducible and $L(\mathbb{C}^n)$ i... | https://mathoverflow.net/users/68327 | Decomposing a reducible representation of the unitary group | Yes, you can do this step by step. It might help to rewrite this as operators acting on $L(\mathbb{C}^n\otimes \mathbb{C}^n)$. Then the tensor product decomposition that you want is $U\otimes U^\ast\otimes U\otimes U^\ast$. Also note that you can restrict to $SU(n)$.
Now, you can use [Littlewood-Richardson](http://en... | 5 | https://mathoverflow.net/users/38947 | 198222 | 95,921 |
https://mathoverflow.net/questions/198229 | 5 | Consider the following graph $G=(V,E)$ where $V=\mathbb{R}^2$ and $E = \{\{x,y\}: x,y \in \mathbb{R}^2 \text{ and } |x-y|\in \mathbb{Q}\}$.
What is $\chi(G)$?
(This is a variant of the [Hadwiger-Nelson problem](http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem).)
| https://mathoverflow.net/users/8628 | A variant to the Hadwiger-Nelson problem | By considering all the rational numbers on the $x$-axis we can see that we need at least countably many colors. This is also sufficient, that is the chromatic number of the rational-distances graph is countable. This is due to Erdos and Hajnal in the case of $\mathbb R^2$. They show that the rational-distances graph in... | 10 | https://mathoverflow.net/users/2384 | 198231 | 95,924 |
https://mathoverflow.net/questions/198232 | 5 | Suppose $B, \{A\_i: i \in \omega\}$ are i.i.d. random variables with uniform distributions on $[0,1]$. If $f$ is a map such that $\{f(A\_i, B): i \in \omega\}$ are independent, must $\{f(A\_i, B): i \in \omega\}$ also be independent of $B$?
| https://mathoverflow.net/users/8106 | Does independence of the sequence $f(A_i, B)$ imply the sequence is independent of $B$? | It is true.
Let $\chi\_Z(x,y)$ be the indicator function of the set $\lbrace (x,y) \,|\, f(x,y)\in Z\rbrace$ for some set $Z$.
The condition that the events $f(A\_1,B)\in Z$ and $f(A\_2,B)\in Z$ are independent is that
$$ \int\_{[0,1]^3} \chi\_Z(x,y)\chi\_Z(x'y)\,dxdx'dy
= \int\_{[0,1]^2} \chi\_Z(x,y) \,dxdy \ \ ... | 5 | https://mathoverflow.net/users/9025 | 198248 | 95,929 |
https://mathoverflow.net/questions/198181 | 0 | Suppose I wish to find the homotopy classes of maps of $B^3 \rightarrow M$ which along the boundary are fixed by a (particular) map $f: S^2 \rightarrow M$. Take $M$ to be a closed orientable $n$-manifold; in my problem $n=9$.
What can I say about the homotopy classes of such maps?
I am sorry if this is a trivial q... | https://mathoverflow.net/users/64341 | Maps of balls with fixed value along boundary | This set of homotopy classes is in bijective correspondence with $\pi\_3(M)$. More generally, let $[B^k,X;f]$ be the set of homotopy classes of maps $B^k\to X$ that restrict to a given $f:S^{k-1}\to X$, where homotopies are also through such maps. The thing to prove is that a homotopy $F:S^{k-1}\times I\to X$ from $f$ ... | 2 | https://mathoverflow.net/users/23571 | 198254 | 95,931 |
https://mathoverflow.net/questions/198255 | 5 | I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an automorphism group?
| https://mathoverflow.net/users/25511 | Structure of the automorphism group of a Riemann surface | There are no obstructions.
In fact, every finite group is isomorphic to the full automorphism group $\textrm{Aut}(S)$ of some compact Riemann surface $S$ (of genus at least $2$).
Moreover, $S$ may be chosen so that the quotient Riemann surface $S/\textrm{Aut}(S)$ has any preassigned genus.
For a reference, see Th... | 12 | https://mathoverflow.net/users/7460 | 198256 | 95,932 |
https://mathoverflow.net/questions/198257 | 2 | Let $(X,\tau)$ be a topological space. We assign to $(X,\tau)$ an equivalence relation $\simeq\_{(X,\tau)}$ in the following way:
>
>
> >
> > $x\simeq\_{(X,\tau)} y$ if and only if there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.
> >
> >
> >
>
>
>
(Reflexivity, symmetry, and transitiv... | https://mathoverflow.net/users/8628 | Equivalence relation defined by the existence of a homeomorphism | I think the following works in quite a lot of situations:
* The equivalence relation $\cong$ induces a partition $\mathcal{P}$.
* Under AC we can well-order $\mathcal{P}$, and for every $P \in \mathcal{P}$, define $U\_{P} = \bigcup\_{Q < P} Q$.
* The $U\_{P}$ form a topology $\tau$, and for all $x,y \in P$, we have $... | 8 | https://mathoverflow.net/users/21815 | 198260 | 95,933 |
https://mathoverflow.net/questions/198036 | 4 | I wonder if someone has proved that $sSet\_{Quillen}$ is not a fibrantly generated model structure ? Do we know something about the possible fibrant generation of $sSet\_{Quillen}$ ?
Thanks
| https://mathoverflow.net/users/nan | fibrant generation of $sSet_{Quillen}$? | sSet is not fibrantly generated. I originally thought the issue would be with the lack of cosmall objects, but it's even worse than that. In fact, **there is no set of maps in sSet that can detect the acyclic cofibrations via lifting**. This fact is due to Bill Dwyer, and I learned it from Remark 5.7 in this excellent ... | 9 | https://mathoverflow.net/users/11540 | 198275 | 95,934 |
https://mathoverflow.net/questions/198282 | 7 | I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise.
Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for eg. a curve) as well as a short exact sequence of smooth connected groups
$1\to K\to G\stackrel{\pi}{\to} H\to 1$
(if o... | https://mathoverflow.net/users/1328 | liftings of principal bundles | The broad outlines of how this business works don't depend on the fact that you're working with varieties so let me work with spaces instead, by which I mean homotopy types.
Let $f : X \to BG$ be a principal bundle and let $BH \to BG$ be a map along which you'd like to lift. Then the space of lifts of $f$ to $BH$ (u... | 6 | https://mathoverflow.net/users/290 | 198286 | 95,938 |
https://mathoverflow.net/questions/198274 | 0 | Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between $h^0(\mathcal{L} \otimes\_{\mathcal{O}\_X} \mathcal{O}\_{V.W})$ and the intersection multiplicity of V.W (like is the former boun... | https://mathoverflow.net/users/54369 | Intersection multiplicty and global sections | If $V$ and $W$ are Cohen--Macaulay, then $V \cap W$ is a finite number of points implies $Tor\_{>0}(O\_V,O\_W) = 0$. This means that $V\cdot W = \ell(O\_V \otimes O\_W) = \ell(L \otimes O\_V \otimes O\_W)$ which is equal to the $h^0$ you are interested in.
| 1 | https://mathoverflow.net/users/4428 | 198291 | 95,940 |
https://mathoverflow.net/questions/198264 | 1 | I searched for it for a long time, but it seems that everybody is taking this for granted and does not bother to point out a proof. Would it be possible that someone points me to a proof or makes me see the obvious.
Thank you very much.
| https://mathoverflow.net/users/6776 | Proof of "generic curve of genus at least 2 has no nontrivial maps to a positive genus curve" | OK, here's an answer. As I said in the comments, if the $C$ is a (smooth projective) curve with a simple Jacobian, then it can't map onto a curve of smaller genus. Since I was a bit curious myself,
I found a reference for the next step: Koizumi "The ring of correspondences on a generic curve of genus g" Nagoya (1976).... | 6 | https://mathoverflow.net/users/4144 | 198296 | 95,944 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.