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/309753 | 3 | The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical. Suppose $G$ is a finite group such that the order of $G$ is not divisible by the characteristic of $K$ and $G$ is acti... | https://mathoverflow.net/users/57804 | Operation of a p'-group on a set of p-power order and fix points | I think the guess in your comment is correct: There is the following result of Glauberman:
>
> **Theorem** (Glauberman). Let the finite group $G$ act on the finite group $N$ by automorphisms, where $(\lvert G \rvert, \lvert N \rvert ) = 1$. Let both $G$ and $N$ on a set $\Omega$, where the action of $N$ is transiti... | 4 | https://mathoverflow.net/users/10266 | 309774 | 134,862 |
https://mathoverflow.net/questions/309726 | 7 | Let $X=(X,d\_X)$ and $Y=(X,d\_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies *Talagrand transportation-cost inequality*, namely
>
> There exists a constant $c\_\mu > 0$ such that
> $$
> W(\nu,\mu) \l... | https://mathoverflow.net/users/78539 | Transportation-cost inequality for pushforward measure | The result you want (but of course with $c\_{\varphi\_\#\mu} \le L^2 c\_\mu$ rather than $c\_{\varphi\_\#\mu} \le L c\_\mu$) is Lemma 2.1 in the paper at <https://arxiv.org/pdf/math/0410172>
| 3 | https://mathoverflow.net/users/36721 | 309775 | 134,863 |
https://mathoverflow.net/questions/309770 | 7 | Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1\_R$.
$\mathbf{A}^1\_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}\_a)\_R$.
$(\mathbf{G}\_a)\_R$ carries a functorial multiplicative action of $R$ making it into an $R$-module object ... | https://mathoverflow.net/users/nan | Vector space objects in schemes - confusion | Briefly, $\mathbb{A}^1\_R$ is not a vector space over $\mathbb{A}^1\_\mathbb{C}$ in a natural way.
Strictly speaking, saying that $\mathbb{A}^1\_R$ is a ring object in schemes in not precise. What is correct is that the morhpism $\mathbb{A}^1\_R\to \mathrm{Spec} R$ (adjoint to $R\to R[T]$) is a ring object in the ca... | 10 | https://mathoverflow.net/users/86006 | 309777 | 134,865 |
https://mathoverflow.net/questions/309771 | 3 | Five simply connected closed 4-manifolds are known to admit Riemannian metrics with nonnegative sectional curvature:
$$\mathbb{S}^4,\,\mathbb{C}\mathbb{P}^2,\,\mathbb{S}^2\times\mathbb{S}^2,\,\mathbb{C}\mathbb{P}^2\#\mathbb{C}\mathbb{P}^2,\,\mathbb{C}\mathbb{P}^2\#\overline{\mathbb{C}\mathbb{P}^2}.$$
Hypothetically, t... | https://mathoverflow.net/users/9833 | Complex surfaces not admitting nonnegative sectional curvature metrics | Gromov proved that there is a constant $C(n)$ such that any complete $n$-manifold $M$ of non-negative curvature satisfies $dimH\_\*(M)\leq C(n)$. Where $C(n)\leq 10^{3n^4+9n^3+6n^2}$. For details reference look at Theorem 3.19 <https://www.math.upenn.edu/~wziller/math660/TopogonovTheorem-Myer.pdf>
And that will give ... | 2 | https://mathoverflow.net/users/33064 | 309779 | 134,866 |
https://mathoverflow.net/questions/309790 | 3 | We know [$H^d(\mathbb{RP}^5,\mathbb{Z}\_2)=\mathbb{Z}\_2$](https://topospaces.subwiki.org/wiki/Cohomology_of_real_projective_space#Coefficients_in_a_module_over_a_2-divisible_ring). So there are two classes of $\mathbb{Z}\_2$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$.
Wha are the Poincaré dual $(5-d)$-d... | https://mathoverflow.net/users/27004 | Poincaré dual of the generators of $H^d(\mathbb{RP}^5,\mathbb{Z}_2)$ | Yes, all of them are true.$\newcommand{\RP}{\mathbb{RP}}\newcommand{\Z}{\mathbb Z}$
First, let's show $\RP^4\subset\RP^5$ is Poincaré dual to $a\in H^1(\RP^5;\Z/2)$. In this case only, there's a nice
geometric shortcut: $a$ determines a principal $\Z/2$-bundle $P\to\RP^5$ with $w\_1(P) = a$, unique up to
isomorphism.... | 4 | https://mathoverflow.net/users/97265 | 309793 | 134,870 |
https://mathoverflow.net/questions/309795 | -5 | In <https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo> T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in Condorcet's paradox, and as a former student in physics, I wonder if this structure could explain why in quantum chromodynami... | https://mathoverflow.net/users/13625 | Borromean rings, Condorcet's paradox and Quantum chromodynamics | [Brunnean links](https://en.wikipedia.org/wiki/Brunnian_link)--in which cutting any component knot leads to the separation of all component knots--exist for any $n>0$ (not just $n=3$ for Borromean links) and Condorcet's paradox also holds for any $n>2$, so any possible relation between them is irrelevant to the specifi... | 7 | https://mathoverflow.net/users/89654 | 309796 | 134,871 |
https://mathoverflow.net/questions/309756 | 11 | If $U$ is a selective ultrafilter on $\omega$, then $U$ generates an ultrafilter in $V^{\mathbb S}$, where ${\mathbb S}$ is Sacks forcing. The same is true with ${\mathbb S}$ being replaced by ${\mathbb S}\_n$, the product of $n$ copies of Sacks forcing, $n<\omega$ (Halpern and Pincus, 1981), and I can see a proof of t... | https://mathoverflow.net/users/114509 | Countable support product of Sacks forcings and selective ultrafilters | The collection of possible large sets is analytic, namely: $\{A\subset \omega: \forall i<\omega\ \exists U\_i\subset T\_i \text{ $U\_i$ is perfect and } f\restriction \bigcup\_{n\in A}\Pi\_{i<\omega} U\_i(n) \text{ is constant}\}$ (here we can assume the length of the roots of $T\_i$ goes to infinity so the coloring $f... | 8 | https://mathoverflow.net/users/23835 | 309800 | 134,873 |
https://mathoverflow.net/questions/309801 | 5 | In one step of solving a difficult problem, I would like to know the largest eigenvalue of a matrix with this pattern:
$$A\_n = \begin{bmatrix}
0 & 0 & 0 & 0 &\dots & 0 \\
0 & 1 & 1 & 1&\dots & 1 \\
0 & 1 &2 &2 &\dots &2\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 1 & 2 & 3 & \dots& n-1
\end{bmatr... | https://mathoverflow.net/users/128503 | Largest Eigenvalue of a Matrix with Special Form in terms of n | Your matrix has entries given by $a\_{ij}=\min(i,j)$, where $0\le i,j\le n-1$. Have a look at Section 3 of [this paper of mine](https://arxiv.org/abs/1411.4107v2) for a derivation of explicit bounds.
| 6 | https://mathoverflow.net/users/8430 | 309802 | 134,874 |
https://mathoverflow.net/questions/309066 | 7 | Many years ago, Grinberg found some uniquely-hamiltonian $3$-connected graphs, and published his results in a paper that has been cited several times as follows.
>
> E. Grinberg, Three-connected graphs with exactly one Hamiltonian cycle, Republican Foundation of Algorithms and Programmes, Computing centre. P. Stuts... | https://mathoverflow.net/users/1492 | Grinberg's uniquely hamiltonian 3-connected graphs (Russian paper) | I have now resolved most of the mysteries, and as MO prompts me to answer my own question, I am now doing so even though it feels a bit odd.
After some false starts with expired email addresses, I managed to contact Dainis Zeps in Latvia, who kindly filled in the missing details.
Basically Zeps and Grinberg were wo... | 5 | https://mathoverflow.net/users/1492 | 309806 | 134,875 |
https://mathoverflow.net/questions/309636 | 3 | Let $f:X\to Y$ be a proper surjective holomorphic map between two $n$-dimensional connected complex manifolds $X$ and $Y$. $X$ is called a proper modification of $Y$ if there are nowhere dense compact analytic subsets $E\subset X$ and $S\subset Y$ such that the following hold:
(1) $f(E)\subset S$.
(2) $f$ maps $X\set... | https://mathoverflow.net/users/128428 | Proper modifications of $\mathbb{C}^{n}$ | No. A counterexample to your question is given by a blow-up at point in $\mathbb C^n$ followed by a blow-up along a compact submanifold contained in the exceptional divisor of the first blow-up.
| 4 | https://mathoverflow.net/users/35428 | 309807 | 134,876 |
https://mathoverflow.net/questions/309785 | 8 | The notion of a stationary set is peculiar in that it applies to subsets of certain very particular posets -- ordinals or powersets. At least to a non-set-theorist, the situation seems to beg for the relevant properties of these posets to be abstracted. I'm wondering if this has been done before. As evidence that it's ... | https://mathoverflow.net/users/2362 | Stationarity and Fodor's lemma for a (nice) poset? | You may look at the paper [Regressive functions and stationary sets](https://doi.org/10.1007/BFb0103112) by Karsten Steffens (In: Müller G.H., Scott D.S. (eds) Higher Set Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), pp. 423–435. Lecture Notes in Mathematics **669** (1978)):
Review [from Mathscinet]... | 13 | https://mathoverflow.net/users/11115 | 309810 | 134,878 |
https://mathoverflow.net/questions/307283 | 4 | If $P(z)$ having no zeros in $|z|<1,$ then
$$\frac{\max\_{|z|=1}|P'(z)|}{\max\_{|z|=1}|P(z)|}\leq \frac{n}{2}.$$
Can we prove this by induction on $n$? or is there any alternative way?
---
Attempt at a proof: let us try to show that the inequality holds by induction on the degree $n$ of the polynomial $P(z)$.
... | https://mathoverflow.net/users/127229 | Induction principle on proving an inequality | The statement is known in the literature as the **Theorem of Erdős and Lax**.
It was conjectured by Erdős and first proved by Lax. Later additional proofs were given by de Bruijn, Aziz--Mohammad, Rahman, and Boas. The proofs of these authors do not use induction.
1. A. Aziz and Q. G. Mohammad: [*Simple Proof of a ... | 5 | https://mathoverflow.net/users/296 | 309827 | 134,883 |
https://mathoverflow.net/questions/309820 | 9 | $\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
\newcommand{\p}{\mathfrak{p}}
$
Let $A$ be an abelian variety over a number field $F$.
It is expected that the $L$-function of $A$ has analytic continuation to $\Bbb C$ and sa... | https://mathoverflow.net/users/84923 | Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties | $\newcommand{\p}{\mathfrak{p}}$By Theorem 6.3 of [this paper by Keith Conrad](http://www.math.uconn.edu/~kconrad/articles/eulerprod.pdf), strong conjectures about $L(A,s)$ (stronger than GRH for this $L$-function, but still "believable"), imply that
$$
\prod\_{N\p\le x}L\_{\p}(A,N\p^{-1}) \sim C (\log x)^r
$$
where $... | 4 | https://mathoverflow.net/users/40821 | 309828 | 134,884 |
https://mathoverflow.net/questions/309494 | 8 | Let $\mathfrak{t}$ be the least ordinal such that $L\_{\mathfrak{t}}$ has undefinable ordinals; i.e. there is an $\alpha<\mathfrak{t}$ such that $L\_{\mathfrak{t}}$ cannot define $\alpha$.
This ordinal is quite large, but may have countable bounds under certain conditions. Because this ordinal is at most $\omega\_1$ ... | https://mathoverflow.net/users/115951 | Is the smallest $L_\alpha$ with undefinable ordinals always countable? | ${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L\_\gamma \prec L\_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L\_{\mathfrak t}$ is $\subseteq$-least with $X \prec L\_{\mathfrak t}$, then $X \not= L\_{\mathfrak t}$; hence if $\sigm... | 12 | https://mathoverflow.net/users/114509 | 309829 | 134,885 |
https://mathoverflow.net/questions/309791 | 9 | For $U\_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra representations of $\frak{g}$. Both categories have an obvious monoidal structure, what is not obvious is if this is an e... | https://mathoverflow.net/users/121660 | Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups | For simplicity let’s just do the $\mathfrak{sl}(2)$ case.
Let X be the 2-dimensional natural representation. From the fusion rules, $\mathrm{Hom}(X \otimes X,1)$ and $\mathrm{Hom}(1, X \otimes X)$ are one-dimensional. Choose a map in each normalized such that the zig-zag is the identity:
$$X = X \otimes 1 \rightarrow X... | 5 | https://mathoverflow.net/users/22 | 309831 | 134,887 |
https://mathoverflow.net/questions/309013 | 21 | Several good references dedicated to hyperbolic groups have been written until 1990, including:
* *Hyperbolic groups*, written by M. Gromov.
* *Géométrie et théorie des groupes : les groupes hyperboliques de Gromov*, written by M. Coornaert, A. Papadopoulos and T. Delzant.
* *Sur les groupes hyperboliques de M. Gromo... | https://mathoverflow.net/users/122026 | Modern references on hyperbolic groups | I think this is a great question, as there is still a need for an authoritative reference about (word-)hyperbolic groups. Since the textbook doesn't exist, I'd like to take the question in a slightly different direction by listing some of the material I think it should cover. (This is inevitably a personal and biased a... | 22 | https://mathoverflow.net/users/1463 | 309841 | 134,892 |
https://mathoverflow.net/questions/309821 | 3 | **Definition.** A closed subset $S$ of a topological space $X$ is called a *separator* between points $x,y\in X\setminus S$ if the points $x$ and $y$ belong to different connected components of $X\setminus S$. A separator $S$ is called an *irreducible* separator between $x$ and $y$ is $S$ coincides with each closed sep... | https://mathoverflow.net/users/61536 | Does each separator between points of a continuum contain an irreducible separator? | No. Consider the subset $X$ of $\mathbb{R}^2$ consisting of the union of line segments beginning at $(0,0)$ and ending at $(1,2^{-n})$ for $n\geq 0$ or $(1,0)$. Let $x=(0,0)$ and $y=(1,0)$ and consider the separator $S$ consisting of points of the form $(\frac{1}{2},2^{-n-1})$ or $(\frac{1}{2},0)$. I claim that $S$ is ... | 3 | https://mathoverflow.net/users/83901 | 309848 | 134,895 |
https://mathoverflow.net/questions/309845 | 2 | Let $(X\_n)\_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
>
> $$
> 0<\lim\_{\beta\rightarrow 1}(1-\beta)\sum\_{n=0}^{\infty}\beta^nX\_n <\infty.
> $$
>
>
>
I have already shown that the limit is finite if $\ma... | https://mathoverflow.net/users/52978 | Divergence rate of geometric sum of random variables | $\newcommand{\be}{\beta}
\newcommand{\E}{\operatorname{\mathsf E}} $
Note that $\mu:=\E X\_1\in(0,\infty]$.
Suppose first that $\mu<\infty$.
Then, for $\be\uparrow1$,
\begin{align\*}
(1-\be)\sum\_{n=0}^{\infty}\be^n X\_n
&=(1-\be)^2\sum\_{n=0}^\infty X\_n\sum\_{j=n}^\infty\be^j \\
& =(1-\be)^2\sum\_{j=0}^\inft... | 1 | https://mathoverflow.net/users/36721 | 309849 | 134,896 |
https://mathoverflow.net/questions/309813 | 4 | Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).
I need a reference to the following facts (which I believe are true at least in dimension $n=3$):
**Fact 1.** For every closed connected subset $A\subset X$ that can be embedded to $\mathbb R^{n-1}$ the complement $X\setminus A$ is connect... | https://mathoverflow.net/users/61536 | The homological negligibility of certain subsets in compact manifolds | Restating my comment above (which linked to another MO post):
For an open subspace $U\subset X$ there is a long exact sequence (via the normal LES for the pair $(X,X-U)$ and excision)
$$ \cdots\to H^\ast\_c(U) \to H^\ast\_c(X) \to H^\ast\_c(X-U) \to H^{\ast+1}\_c(U)\to\cdots$$
and Poincaré duality with compact suppor... | 3 | https://mathoverflow.net/users/12310 | 309854 | 134,899 |
https://mathoverflow.net/questions/309855 | 2 | Let $A$ be a $\sigma$-unital $C^\*$-algebra and $A\_s:=A\otimes K$ its stabilization (where $K$ is the algebra of compact operators on a separable Hilbert space). Is it true that there exist an approximation of unity $P\_n\in A\_s$ with $P^\*\_n=P\_n=P\_n^2$, in general?
| https://mathoverflow.net/users/nan | Approximation of unity by projectors | What if $A=C\_0([0,1))$, the continuous functions $f:[0,1]\rightarrow\mathbb C$ with $f(1)=0$? Then $A\otimes K = C\_0([0,1), K)$ the space of norm continuous $f:[0,1]\rightarrow K$ with $f(1)=0$.
Then, if $f=f^\*=f^2$ then $f(s)=f(s)^\* = f(s)^2$ for each $s\in [0,1]$. Thus $f(s)$ is a projection for each $s$, so $\... | 3 | https://mathoverflow.net/users/406 | 309866 | 134,903 |
https://mathoverflow.net/questions/309852 | 3 | I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant literature.
Consider a random variable $X$ distributed hypergeometrically with parameters $(n,m,i)$, i.e.,
$$p\_X(x)=\fra... | https://mathoverflow.net/users/128529 | Lower bound on the sum of pmf squared of a hypergeometric distribution | Suppose that $i=n-i=m$ (so that $n=2m$) and $m$ is even; I suppose that such $i$ and $m$ you consider relevant. Then
\begin{equation}
p\_X(x)=\binom mx^2\Big/ \binom{2m}m\le\binom m{m/2}^2\Big/ \binom{2m}m
\asymp\frac{(2^m/\sqrt m)^2}{2^{2m}/\sqrt m}=\frac1{\sqrt m}\asymp\frac1{\sqrt n},
\end{equation}
whence
\beg... | 3 | https://mathoverflow.net/users/36721 | 309871 | 134,905 |
https://mathoverflow.net/questions/309837 | 5 | Let $S$ be some base scheme, $H$ a finite flat group scheme over $S$, and $\alpha: \mu\_p \to H$ a homomorphism of group schemes ($p$ a prime). Is the kernel of $\alpha$ necessarily flat over $S$?
(I know that kernels of general homomorphisms of FFGS $G \to H$ need not be flat, but I don't know of a counterexample wh... | https://mathoverflow.net/users/2481 | Kernels of homomorphisms of group schemes | This holds for any homomorphism $f: G\to H$ with $G$ of multiplicative type and of finite type, and $H$ separated and finitely presented. Here I assume that by "finite flat" you mean "finite locally free".
Reference: SGA3, IX, Thm 6.8.
| 8 | https://mathoverflow.net/users/7666 | 309872 | 134,906 |
https://mathoverflow.net/questions/309228 | 7 | Consider the real diagonal $4\times 4$ - matrix
$$I\_{2,2}={\rm diag}(1,1,-1,-1)$$
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I\_{2,2}\cdot \bar g ^{\rm tr}=I\_{2,2}\}.$$
We regard $G$ as an algebraic group over ${\mathbb{R}}$.
It is known that $G$ is q... | https://mathoverflow.net/users/4149 | Explicit description of SU(2,2)/U | I guess your variety is just the variety of pairs of ${\mathbb C}$-linearly independent vectors in ${\mathbb C}^4$ that are isotropic with respect to this Hermitian form and orthogonal to each other. Respectively, the twisted form is the same variety but for another Hermitian form (and if the form is not hyperbolic, th... | 4 | https://mathoverflow.net/users/5107 | 309874 | 134,907 |
https://mathoverflow.net/questions/309875 | 8 | Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. The absolute galois group $G\_\mathbb{Q}$ of $\mathbb{Q}$ acts on the set of real-closed subfields of $\overline{\mathbb{Q}}$.
>
> Does it act transitively?
>
>
>
The real-closed subfields are in bijection with the involutions of $G\_\mathbb... | https://mathoverflow.net/users/12419 | Are all real-closed subfields of $\overline{\mathbb{Q}}$ conjugate? | Any real-closed subfield $R\subseteq\overline{\mathbb Q}$ is a real closure of $\mathbb Q$ (being real closed and algebraic over $\mathbb Q$). Thus, by uniqueness of real closures, any two such fields are isomorphic, and an isomorphism of $R$ to $R'$ extends to an isomorphism of $R(i)=\overline{\mathbb Q}$ to $R'(i)=\o... | 12 | https://mathoverflow.net/users/12705 | 309878 | 134,908 |
https://mathoverflow.net/questions/309747 | 4 | I want to solve the following non-linear matrix equation for $X\in\mathbb{R}^{N\times N}$:
\begin{equation}
XX^{\top}+ABX^{\top}-A=0 \qquad (1)
\end{equation}
For a given matrices $A\in\mathbb{R}^{N\times N}, B\in\mathbb{R}^{N\times N}$ and $A=A^{T}, A\succeq0$.
Is there a known stable numerical solution for (1)... | https://mathoverflow.net/users/128477 | Non-linear matrix equation | I'll give an explicit expression for a family of solutions to your first problem (the more general one without the symmetry constraint).
Let us use Robert Israel's suggestion as in your last edit, and start from
\begin{equation}
XX^{\top}+\frac{1}{2}ABX^{\top}+\frac{1}{2}XB^{\top}A-A=0 \tag{4}
\end{equation}
You c... | 6 | https://mathoverflow.net/users/1898 | 309879 | 134,909 |
https://mathoverflow.net/questions/309718 | 1 | Let $Z = X\cap Y\subset\mathbb{C}^N$ be a manifold given as the intersection of two manifolds $X,Y$ intersecting transversally along $Z$. Let $T\_p^kX,T\_p^kY,T\_p^kZ$ be the $k$-osculating spaces at $p\in Z$ of $X,Y,Z$ respectively.
Is it true that $T^k\_pZ = T^k\_pX\cap T^k\_pY$? I know the answer is positive for $... | https://mathoverflow.net/users/nan | Osculating spaces of intersection of two varieties | No, it does not hold for higher order osculating spaces.
In the projective space $\mathbb{P}^{5}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^3$, and let $C\subset\mathbb{P}^3$ be a degree $3$ rational normal curve. Fixed an isomorphism $\psi:\mathbb{P}^1\rightarrow C$ we consider the rational norma... | 0 | https://mathoverflow.net/users/14514 | 309885 | 134,911 |
https://mathoverflow.net/questions/309876 | 2 | The idea to construct a heat kernel is first construct a parametric in a small neighbourhood. Then use a bump function to extend it. And do convolution iteratively. (Reference: Laplacian on a Riemannian manifold [ROSENBERG].)
My question is will it spread all over the whole manifold? It seems that the bump function c... | https://mathoverflow.net/users/90295 | Heat kernel on Riemannian manifold | This bump function is not supported on an arbitrary neighborhood, but one very specific to the construction of $H(t,x,y)$. Rosenberg's argument proceeds by constructing the parametrix $H\_k(t,x,y)$ in a neighborhood of the diagonal $M\_{\text{diag}}\subset M\times M$. Namely the neighborhood $U\_\epsilon=\{(x,y)\in M\t... | 3 | https://mathoverflow.net/users/111338 | 309887 | 134,912 |
https://mathoverflow.net/questions/309710 | 13 | Let $\square=[0,1]\times[0,1]$ be the unit square
and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary.
Assume $f$ is a limit of homeomorphisms $\square\to \square$. (By [Moore's theorem](https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9C%D1%83%D1%80%D... | https://mathoverflow.net/users/1441 | Limit of homeomorphisms from square to square | Steve Ferry gave me an answer --- the answer is "no".
In fact according to "A continuous decomposition of the plane into pseudo-arcs" by Wayne Lewis and John Walsh,
there is a continuous subdivision of the plane into [pseudo-arcs](https://en.wikipedia.org/wiki/Pseudo-arc).
My Moore's theorem, the quotient space is al... | 6 | https://mathoverflow.net/users/1441 | 309889 | 134,913 |
https://mathoverflow.net/questions/309883 | 2 | Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle\_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $(,)$ on $Y$. Take $y \in Y$, and define the map
$l\_y : x' \in X \mapsto (A x', y)$.
Assume this last map is bounded, ... | https://mathoverflow.net/users/128536 | Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$ | Consider $H = L^2[0,\infty)$ with $(,) = \langle,\rangle$, $A: H \to H$ the shift operator $A f(t) = f(t+1)$, so that
$$ A^\* f(t) = \cases{f(t-1) & if $t \ge 1$\cr
0 & otherwise\cr} $$
Then $A$ is not Fredholm, but $A A^\* = I$.
| 8 | https://mathoverflow.net/users/13650 | 309891 | 134,915 |
https://mathoverflow.net/questions/309590 | 4 | I am reading the book "The Arithmetic of Hyperbolic Three Manifolds" by Maclachlan and Reid and I am having some problems in understanding something about cusps.
The definition they give of a cusp is the following (1.2.7):
A point $\zeta \in \overline{\mathbb{C}}$, the sphere at infinity, is a cusp of the Kleinian ... | https://mathoverflow.net/users/128408 | Cusps in hyperbolic manifolds and fundamental group | This is true for sufficiently small Margulis constant (depending on the manifold). Just make it smaller than the translation length in the smallest Margulis tube.
For the second, there will be a horoball invariant under the action of each cusp. Shrink the horoball until the minimal translation length of a parabolic ... | 4 | https://mathoverflow.net/users/1345 | 309900 | 134,919 |
https://mathoverflow.net/questions/309901 | 5 | Let $X$ be an infinite set, and let $(A\_n)\_{n\in\omega}$ be a collection of subsets of $X$ with the following properties:
1. $|A\_m\cap A\_n| \leq 1$ for $m\neq n\in \omega$, and
2. $|A\_n|=\aleph\_0$ for all $n\in \omega$.
We consider the following statement:
>
> (EFL$\_\omega$:) There is $f:X\to \omega$ suc... | https://mathoverflow.net/users/8628 | Countable version of Erdös-Lovasz-Faber conjecture | If I understand it correctly, it's false. Let $x \notin A\_0 = \{1,2,\dots \}$. Then let $A\_i$ all meet at $x$, and also each meet $A$ at $i$ (add extra elements as necessary; they should be irrelevant). Then $f(x) \neq f(i)$ for any $i$, so $f(x) \notin f(A)$.
This didn't work in the finite case because the sets me... | 7 | https://mathoverflow.net/users/44191 | 309904 | 134,921 |
https://mathoverflow.net/questions/309907 | 8 | Let $X$ be a $k$-connected spectrum for $k \in \Bbb{Z}$.
I want to deduce how connected the counit of $(\Sigma^\infty, \Omega^\infty)$- adjunction is, that is, how connected is the map
$$
\Sigma^\infty\Omega^\infty X \to X.
$$
Any help would be appreciated.
| https://mathoverflow.net/users/117088 | Connectivity of suspension-loop adjunction | If the spectrum $X$ is $r$-connected, then the map $\Sigma^\infty\Omega^\infty X \to X$
is $(2r+2)$-connected.
Here's a sketch: apply the functor $\Omega^\infty$ to get the map of spaces
$$
Q(\Omega^\infty X) \to \Omega^\infty X
$$
where $Q = \Omega^\infty\Sigma^\infty$.
It will be enough to identify the connectivity... | 11 | https://mathoverflow.net/users/8032 | 309913 | 134,924 |
https://mathoverflow.net/questions/309614 | 4 | For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M\_{\phi}:=Y\bigcup\_{\phi} (X \times \{ 1\})$. Denote $\pi\_n (M\_{\phi},X \times \{ 1\} )$ by $\pi\_n (\phi)$. The map $\phi$ is called $n$-connected if $X$ and $Y$ are connected and $\pi\_i (\phi)=0$ for $1\leq i\leq n... | https://mathoverflow.net/users/114476 | A question about Wall's construction for CW-complexes | The given conditions imply that the map $\psi:L\rightarrow X$ induces an isomorphism of fundamental groups, and so it lifts to a map $\tilde\psi: \widetilde L\rightarrow \widetilde X$. The conditions also imply that $\tilde\psi$ is an isomorphism on homology. Since these spaces are simply-connected, their homology grou... | 2 | https://mathoverflow.net/users/124004 | 309914 | 134,925 |
https://mathoverflow.net/questions/237235 | 9 | A well known result (stated and credited to Todorcevic in "Semiselective Coideals", by Farah, Mathematika, 1997, but with antecedents going back to Mathias) says that, under the appropriate large cardinal hypothesis (enough to get all sets of reals in $L(\mathbb{R})$ to be universally Baire, say), a selective ultrafilt... | https://mathoverflow.net/users/16107 | Nonexistence of generic objects over $L(\mathbb{R})$ | The book draft linked to below shows that existence of a weakly compact Woodin cardinal implies the existence of $L(\mathbb{R})$-generic filters for the following partial orders (all ordered by containment) : (1) the partial order of countable injections from $\mathbb{R}$ to $\mathbb{R}$; (2) the partial order of count... | 6 | https://mathoverflow.net/users/31807 | 309918 | 134,927 |
https://mathoverflow.net/questions/309925 | 4 | For a Weyl group $W$, I would like to know whether each $w\in W$ can be expressed as $w=s\_{\alpha\_1}s\_{\alpha\_2}\cdots s\_{\alpha\_k}$ for some distinct positive roots $\{\alpha\_1, \alpha\_2, \cdots, \alpha\_k\}\subseteq \Phi^+$.
I know for type A, the above is true.
Since $W(A\_n)\cong S\_{n+1}$ with the map... | https://mathoverflow.net/users/110229 | Each $w\in W$ can be expressed as product of distinct reflections? | The answer is "yes" and there is a geometric explanation.
Let $\mathcal{H}$ denote the set of hyperplanes corresponding to the reflections $s\_\alpha$ with $\alpha\in\Phi^+$ (note that $s\_\alpha=s\_{-\alpha}$), and let $\Sigma$ denote the connected components of $V\setminus\bigcup\_{H\in\mathcal{H}}H$ (where $V$ is ... | 8 | https://mathoverflow.net/users/86006 | 309927 | 134,929 |
https://mathoverflow.net/questions/309921 | 4 | Dedekind proved that the free modular lattice on 3 generators is realisable by the intersections and sums of 4-dimensional subspaces in 8-space. Birkhoff showed that the free lattice is infinite if it has at least 4 generators. My questions:
1. Is every free modular lattice realisable by subspaces in a vector space?
... | https://mathoverflow.net/users/10481 | Is the free modular lattice linear? | The answer to the first question is no. Bjarni Jónsson proved that lattices of modules must satisfy the Arguesian identity. This is the lattice theoretic analogue of Desargues' theorem in projective geometry. You can find a discussion of the result in [this survey article](http://www.math.hawaii.edu/~jb/bjarni_jonsson_... | 5 | https://mathoverflow.net/users/3711 | 309929 | 134,930 |
https://mathoverflow.net/questions/309896 | 3 | I was messing around with the intuition behind the size of weakly compact cardinals (in their usual characterization). I found an interesting, seemingly weaker LCA which still implies weak inaccessibility.
---
I started by making an intuitively powerful property that can be stated as an $\mathcal{L}\_{\kappa,\kap... | https://mathoverflow.net/users/115951 | A weakening of cardinal compactness - is it equivalent? | **Theorem:** If $\kappa$ is weakly Skolem then the tree property holds at $\kappa$.
**Proof:** let $\mathcal T$ be a $\kappa$-tree. Let us define two sequences of constants $\langle d\_\alpha \mid \alpha < \kappa\rangle$ and $\langle d\_x \mid x \in \mathcal T\rangle$. Let us consider the theory $T$ with the followi... | 4 | https://mathoverflow.net/users/41953 | 309937 | 134,932 |
https://mathoverflow.net/questions/309945 | 34 | [OEIS sequence A210247](https://oeis.org/A210247) gives the signs of $\text{li}(-n,-1/3) = \sum\_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$.
Mikhail Kurkov noticed that it appeared that $a(n+28) = -a(n)$ for this sequence.
It's not quite true: the first $n$'s for... | https://mathoverflow.net/users/13650 | A remarkable almost-identity | Consider $F(z) = 4/(3+\exp(4z))$ as a function of the complex variable $z$. It is meromorphic and has simple poles where the denominator vanishes. Namely when $4z = \log 3 + (2k +1)\pi i$ for integers $k$. The poles with the smallest magnitude of $z$ occur when $4z = \log 3 \pm \pi i$. We can compute the Taylor series ... | 53 | https://mathoverflow.net/users/38624 | 309948 | 134,935 |
https://mathoverflow.net/questions/309966 | 3 | I have two problems related to eigenvalues of negative definite matrices:
1. I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M\_{11}-M\_{12}M\_{22}^{-1}M\_{21}$ by taking $M=[M\_{ij}]$. Now I want to derive the relation between the eigenvalues of $M$ and $S$. I am particularly looking f... | https://mathoverflow.net/users/128364 | What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement? | As for your first question, $S^{-1}$ is a diagonal block of $M^{-1}$. You can only say that its eigenvalues are interlaced with those of $M^{-1}$. Since all the eigenvalues of $M$ and $S$ are negative, it amounts to saying that the eigenvalues of $S$ are interlaced with those of $M$~:
$$\lambda\_1\le\mu\_1\le\lambda\_2... | 3 | https://mathoverflow.net/users/8799 | 309969 | 134,941 |
https://mathoverflow.net/questions/309899 | 2 | Consider the polynomial ring $R=\mathbb C[x\_1,x\_2,...,x\_{16}]$, and set
$$X=\begin{pmatrix} x\_1 &x\_2&x\_3 &x\_4\\ x\_5&x\_6& x\_7&x\_8\\x\_9&x\_{10}&x\_{11}&x\_{12}\\x\_{13}&x\_{14}&x\_{15}&x\_{16}\end{pmatrix}.$$
Now, using these three matrices
$$L=\begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0 \end{p... | https://mathoverflow.net/users/127118 | radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices | First, let us do all the calculations over $\mathbb{R}$ onstead of over $\mathbb{C}$. It will facilitate things, and will not change the result.
Let us rewrite the equations a bit:
You have $$X\cdot (LX^tL^t)=Id$$ and similarly for $M$ and $N$.
In particular, your system is now equivalent to the sent of equations;
... | 3 | https://mathoverflow.net/users/41644 | 309984 | 134,947 |
https://mathoverflow.net/questions/309981 | 3 | Let $N$ be an even integer, $N>2$. Let $E$ be the set of all *outer* automorphisms $\phi$ of $G = SU(N)$ which are of order 2, i.e. $\phi \circ \phi = \mathrm{id}\_G$.
Choose a particular element $\psi \in E$. Since $\mathrm{Out}(G) \simeq \mathbb{Z}\_2$, for all $\phi \in E$ there exists a matrix $A\_{\phi} \in G$ ... | https://mathoverflow.net/users/61018 | A partition of the set of order 2 outer automorphisms of $SU(N)$ | I don't know that the partition has a name, so to speak, but it is well-understood and falls into the classification of the symmetric spaces of type A. Namely, those of type AI, which are $\mathrm{SU}(n)/\mathrm{SO}(n)$ and, when $n=2m>2$ is even, those of type AII, which are $\mathrm{SU}(2m)/\mathrm{Sp}(m)$.
$\mathr... | 3 | https://mathoverflow.net/users/13972 | 309986 | 134,948 |
https://mathoverflow.net/questions/309931 | 15 | Let $X$ be a complex irreducible variety and denote its smooth locus by $X^{smooth}$. I would like to know what can be said about the induced maps $H\_i(X^{smooth};\mathbb{Q})\rightarrow H\_i(X;\mathbb{Q})$ for $i$ small when the codimension of the singular locus is large. Of course, since the ambient space $X$ may not... | https://mathoverflow.net/users/128556 | The homology groups of the smooth locus of a singular variety | I am adding some additional details to the comment above, since somebody else asked me about this recently. Results about extensions of cohomology classes to all of $X$ from an open subset $U=X\setminus Z$ (or dually, proving that homology classes are obtained by pushforward from an open subset) are usually called *Pur... | 14 | https://mathoverflow.net/users/13265 | 309990 | 134,950 |
https://mathoverflow.net/questions/309985 | 4 | **Question.** If $A\subset \mathbb{R}^n$ is any set of positive Lebesgue $n$-measure, does there exists a Lipschitz map $f:A\to\mathbb{R}^n$ such that $f(A)$ is a ball with the same measure?
In dimension $n=1$, such a map is easily found by defining $f(x) = \mathcal{L}^1(A\cap [0,x])$. In higher dimensions the proble... | https://mathoverflow.net/users/91774 | Existence of a Lipschitz map from a positive measure set to a ball | The problem is also mentioned in
>
> Alberti, Giovanni and Csörnyei, Marianna and Preiss, David: *Structure of null sets in the plane and applications*. European Congress of Mathematics, 3–22, Eur. Math. Soc., Zürich, 2005. [MR2185733](https://mathscinet.ams.org/mathscinet-getitem?mr=2185733). [PDF](http://pagine.... | 2 | https://mathoverflow.net/users/90407 | 309991 | 134,951 |
https://mathoverflow.net/questions/309902 | 3 | Let $D$ be a $2$ dimensional distribution of $\mathbb{R}^3$.
Is there a $1$ dimensional foliation of $\mathbb{R}^3$ with Frenet curves such that for every leaf $\gamma$ of the foliation we have $\mathrm{span}(\gamma'(t), \gamma ''(t))=D(\gamma(t))$ where $\gamma(t)$ is the unit speed parametrization of the leaf $\gamma... | https://mathoverflow.net/users/36688 | Can we foliate the space $\mathbb{R}^3$ with Frenet curves whose tangent and normal vectores span a given $2$ dimensional distribution? | The answer is 'no' in general for an arbitrary Riemannian metric $g$ on a $3$-manifold $M$ and $2$-plane field $D\subset TM$.
I'll give the argument for the flat metric on $\mathbb{R}^3$ and leave the (easy) generalization to arbitrary metrics for the interested.
Let $M$ be $\mathbb{R}^3$ endowed with the flat metr... | 8 | https://mathoverflow.net/users/13972 | 309995 | 134,953 |
https://mathoverflow.net/questions/309964 | 17 | Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$
superconformal field theories ([Wikipedia](https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory), [nLab](https://ncatlab.org/nlab/show/6d+%282%2C0%29-superconformal+QFT), [PhysicsOverflow](https:... | https://mathoverflow.net/users/97265 | What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT? | If you take the (2,0) theory and put it on a manifold which is $T^2 \times M\_4$, it is known to reduce to $\mathcal{N} = 4$ super-Yang Mills theory on $M\_4$. That theory exhibits S-duality, which has been shown to be related to geometric Langlands. In particular, S-duality is part of an $SL(2,\mathbb{Z})$ symmetry, a... | 11 | https://mathoverflow.net/users/947 | 309998 | 134,955 |
https://mathoverflow.net/questions/309997 | 3 | Let $u\_0 \in \dot{H}^{1/2}(\mathbb{R}^3)$. The Fujita-Kato theorem gives rise to a local unique solution $(t,x) \mapsto u(t,x)$ to the Navier-Stokes equations $$\left\{
\begin{array}{ccc}
\partial \_t u + u\cdot \nabla u- \Delta u + \nabla p&=&0\\
div \;u&=&0 \\
u(t=0)&=&u\_0.
\end{array}
\right.$$
The Fujita-Kato ass... | https://mathoverflow.net/users/94415 | A solution to the Navier-Stokes equation that is defined for on $[0,T]$ with $T$ large is global? | $\dot{H}^{1/2}$ is critical with respect to scaling.
Let $\tilde{u}(t,x) = \lambda u(\lambda^2 t, \lambda x)$. Then $\tilde{u}$ solves the Navier-Stokes equation up to $\tilde{T}^\* = T^\* \lambda^{-2}$, with pressure $\tilde{p} = \lambda^2 p(\lambda^2 t, \lambda x)$.
One can check that the corresponding initial ... | 6 | https://mathoverflow.net/users/3948 | 310011 | 134,961 |
https://mathoverflow.net/questions/310020 | 6 | Consider the [Bernoulli numbers](http://mathworld.wolfram.com/BernoulliNumber.html) denoted by $B\_n$, which are rational numbers.
It is known that the harmonic numbers $H\_n=\sum\_{k=1}^n\frac1k$ are not integers once $n>1$.
I am curious about the following:
>
> **Question:** If $n>0$, will $\sum\_{k=0}^nB\_k... | https://mathoverflow.net/users/66131 | Summing Bernoulli numbers | It can never be an integer for $n>0$. There is a result by K.G.C. von Staudt and independently by T. Clausen that
$$B\_n+\sum\_{p\in \mathbb{P}\, ,\, p-1|n}\frac{1}{p}\in \mathbb Z$$
>
> [1] T. Clausen. Lehrsatz aus einer Abhandlung uber die Bernoullischen Zahlen.
> Astr. Nachr., 17:351–352, 1840
>
>
> [2] K. G.... | 12 | https://mathoverflow.net/users/2384 | 310024 | 134,967 |
https://mathoverflow.net/questions/310016 | 7 | According to [Cantor's attic](http://cantorsattic.info/Vopenka#Strong_Compactness_of_Logics), Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any "logic". But I can't find a definition of what a "logic" is either there or in any of the cited references.
**Question:** What is a ... | https://mathoverflow.net/users/2362 | Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any "logic"? | ORIGINAL RESPONSE:
==================
<https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents> Is the article where it is from. It seems to have never been added to the library, which would be my fault.
<https://projecteuclid.org/download/pdf_1/euclid.pl/1235417266> is the first chapter of the textbook w... | 5 | https://mathoverflow.net/users/115951 | 310046 | 134,980 |
https://mathoverflow.net/questions/309935 | 0 | If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X\_0, \dots, X\_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I\_m $, the $m$ -th graded piece of $I$ for every $m$, the quotient ring $ \mathbb{C} [X\_0, \dots, X\_n]/I $ is a regular ring?
Acc... | https://mathoverflow.net/users/89900 | Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring? | $\mathbb{C}[x\_1,\dotsc,x\_n]/I$ is regular if and only if the affine variety $V(I)$ is smooth. When $I$ is homogeneous,a $V(I)$ is a cone (with vertex at the origin). The only way for a cone to be smooth is if it's a linear subspace. So, for homogeneous $I$, the ring is regular if and only if $I$ is generated by linea... | 2 | https://mathoverflow.net/users/88133 | 310051 | 134,982 |
https://mathoverflow.net/questions/309884 | 9 | Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = R\Gamma(C,df:T\_C\to f^\*T\_X)$. Explicitly, this complex may be realized using the Dolbeault resolution of $T\_C$ and $... | https://mathoverflow.net/users/110236 | DGLA controlling deformation of holomorphic curves | Firstly, I assume you mean deformations of $C$ over $X$ ("deformations of $f$" is ambiguous, as it could mean fixing neither or both of $C$ and $X$).
The DGLA philosophy is then that there should exist some DGLA quasi-isomorphic to the explicit realisation of the complex $L$ you wrote down. It doesn't guarantee a DGL... | 10 | https://mathoverflow.net/users/103678 | 310055 | 134,983 |
https://mathoverflow.net/questions/310061 | 7 |
>
> **Question 1.** Let $a,b>1$ be two natural numbers. Is there a prime number $p\in 1+b\mathbb N$ such that $a+p\mathbb Z$ is a generator of the multiplicative group of the field $\mathbb Z/p\mathbb Z$?
>
>
>
In need this fact for establishing some properties of the Golomb topology on positive integers.
... | https://mathoverflow.net/users/61536 | A stronger form of the Dirichlet Theorem on prime numbers in arithmetic sequences | This is a hybrid of Dirichlet's theorem with [Artin's conjecture on primitive roots](https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots). Artin's primitive root conjecture says that if $a \in \mathbf{Z}$ is not a perfect square or $-1$, then there are infinitely many p such that a is a generator mod ... | 16 | https://mathoverflow.net/users/2481 | 310065 | 134,987 |
https://mathoverflow.net/questions/310083 | 4 | I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.
The stacks project <https://stacks.math.columbia.edu/tag/02IN> says that a scheme (a curve in particular) $X$ is Cohen Macaulay if for every $x \in X$, there is an open subset $U$ of $X$ containing $x$ such that the mo... | https://mathoverflow.net/users/43027 | difference between Cohen Macaulay and locally Cohen Macaulay curve | Following the usual meaning of "local" in Algebraic Geometry, a locally noetherian scheme is locally Cohen-Macaulay if and only if $\mathcal{O}\_{X, \, x}$ is a Cohen-Macaulay local ring for every $x \in X$.
This is actually equivalent to your definition of Cohen-Macaulay scheme, see [The Stacks Project, Lemma 27.8.2... | 3 | https://mathoverflow.net/users/7460 | 310085 | 134,991 |
https://mathoverflow.net/questions/310078 | 1 | I have the following basic question. Everything is over $\mathbb{C}$.
Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow Y$. By this I mean that f is birational and surjective (e.g. induced by a big and semiample line bundle) and the excep... | https://mathoverflow.net/users/124888 | Small contraction for Hyperkähler Varieties | Let $f:X\to Y$ be a birational contraction where $X$ is hyperkähler, then $K\_X\sim 0$ and $K\_Y=f\_\*K\_X\sim 0$, and hence $K\_X=f^\*K\_Y$. In particular, this means that $Y$ has canonical singularities. Hence by a result of Hacon and Mckernan [On Shokurov's rational connectedness conjecture, Duke Math. J.
Volume 138... | 2 | https://mathoverflow.net/users/42636 | 310088 | 134,993 |
https://mathoverflow.net/questions/309759 | 8 | If $(X,\tau)$ is a topological space, let $FH(X)$ denote the collection of $x\in X$ such that there is a non-identity homeomorphism $\varphi:X\to X$ with $\varphi(x) = x$.
What is an example of a $T\_2$-space $(X,\tau)$ such that $FH(X)$ is dense in $X$, but $FH(X)\neq X$?
| https://mathoverflow.net/users/8628 | Set of homeomorphic fixed points that is dense, but not equal to whole space | Such an example can be constructed unifying two pathological examples of Cook and van Mill.
**Example** ([Cook, 1967](http://matwbn.icm.edu.pl/ksiazki/fm/fm60/fm60123.pdf)): There exists a non-degenerated metric continuum $K$ such that any continuous map $f:K\to K$ is either constant or the identity.
**Example** ([... | 4 | https://mathoverflow.net/users/61536 | 310098 | 134,998 |
https://mathoverflow.net/questions/309992 | 7 | Consider the following construction: Define $G\_n$ to be the contractible groupoid on $n+1$ objects. Choosing a linear order on the objects of each $G\_n$ turns $G\_\*$ into a cosimplicial object. Define the cosimplicial simplicial set $J\_\*=N(G\_\*)$.
This cosimplicial object defines a Quillen pair between the Quil... | https://mathoverflow.net/users/1353 | Direct comparison from the Rezk hom to the hom of a simplicial category along the coherent nerve? | Edit: The proof in the original answer below the line is correct, but it doesn't prove everything we need to establish to prove the equivalence between qCat and sCat. It turna out this was all worked out in great detail in the paper [Mapping Spaces in Quasicategories](https://arxiv.org/abs/0911.0469) by Dugger and Spiv... | 1 | https://mathoverflow.net/users/1353 | 310117 | 135,004 |
https://mathoverflow.net/questions/310086 | 1 | I'm looking for the residues of the following function $$s \mapsto\sum^\infty\_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$
at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive number.
Yet, I have literally no idea how to precedure here. I tried to rearrange the terms in order to get some wel... | https://mathoverflow.net/users/128226 | Residues of Zeta-like Function | The following is just a sketch, for the detail you can find in the reference of Zagier [<http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValeursZeta/ZetaFunctionRQF.pdf]>. Let us denote
$$f(x,y)=(x+y)e^{-(axy+(x-y)^2)}.$$
Then you can check that
$$F(s):=\sum\_{m,n\ge 1}\frac{m+n}{(amn+(m-n)^2)^s}=\frac{1}{\Gamma(s... | 2 | https://mathoverflow.net/users/110368 | 310118 | 135,005 |
https://mathoverflow.net/questions/310104 | 3 | Is it possible to turn an inaccessible cardinal in $V$ to a successor of a singular cardinal in some forcing extension?
| https://mathoverflow.net/users/119731 | Collapse an inaccessible cardinal to a successor of a singular cardinal | There are several ways to do it:
One is suggested by Noah in his comment.
Another one is to use the supercompact extender based Prikry forcing of Merimovich. See [Supercompact extender based Prikry forcing](http://www2.mta.ac.il/~carmi/Publications/Merimovich%202011%20Supercompact%20extender%20based%20Prikry%20forc... | 4 | https://mathoverflow.net/users/11115 | 310123 | 135,006 |
https://mathoverflow.net/questions/310130 | 6 | By [Grunwald-Wang Theorem](https://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem), if for some odd number $n$ the equation $x^n=a$ has no solutions in $\mathbb Z$, then the equation $x^n=a\mod p$ has no solutions for some prime number $p$. I am interested if we can always choose $p$ is the arithmetic sequence $1+... | https://mathoverflow.net/users/61536 | A simultaneous generalization of the Grunwald-Wang and Dirichlet Theorems on primes | $\newcommand{\Z}{\mathbf{Z}}$
$\newcommand{\Q}{\mathbf{Q}}$
$\newcommand{\F}{\mathbf{F}}$
$\newcommand{\OK}{\mathcal{O}\_K}$
**EDIT**. To prove the existence of at least one prime (or infinitely many primes) meeting the OP's requirement, there is a much simpler argument, see the answer by a so-called friend Don. My a... | 5 | https://mathoverflow.net/users/6506 | 310148 | 135,015 |
https://mathoverflow.net/questions/310152 | 3 | The statement I am familiar with regarding classification of vector bundles is :
>
> Given a paracompact space $X$. The set of isomorphism classes of rank $n$ vector bundles over $X$ is in bijective correspondence with the set $[X,G\_n]$ of homotopy classes of maps from $X$ to $G\_n$.
>
>
>
I am more or less c... | https://mathoverflow.net/users/118688 | Motivation for classifying vector bundles | At Praphulla Koushik's request I am posting my comments above as an answer, with a little extra detail added.
Complex line bundles are classified up to isomorphism by their first Chern class. To see this, consider the long exact sequence of cohomology associated to the exponential sequence
$$0\rightarrow \mathbb{... | 2 | https://mathoverflow.net/users/98320 | 310158 | 135,018 |
https://mathoverflow.net/questions/310124 | 3 | For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we can define the Verma module $\text{Ind}\_{\mathfrak b}^{\mathfrak g} \mathbb C\_{\lambda}$, which is the induced module ... | https://mathoverflow.net/users/75041 | Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras | This line of questioning has been pursued in greater generality. starting in prime characteristic by Ron Irving (and myself) and then in the analogous setting of category $\mathcal{O}$ for a semisimple Lie algebra over $\mathbb{C}$ [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=1231714) . Refinements involvi... | 1 | https://mathoverflow.net/users/4231 | 310159 | 135,019 |
https://mathoverflow.net/questions/310163 | 13 | Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N\_0$ where $\mathbb N\_0=\{0\}\cup\mathbb N$, where $a,b\in\mathbb N$. This topology is often referred to as the *Furstenberg topology* or the *profinite topology*. The space ... | https://mathoverflow.net/users/61536 | Is the identity function a unique multiplicative homeomorphism of $\mathbb N$? | No.
First observe that the automorphisms of the semigroup $\mathbf{N}^\*$ (which you denote $\mathbb{N}$) are induced by permutations of primes.
Consider the automorphism $f$ induced by the transposition $(2,3)$ (thus, mapping $2^a.3^b.c$ to $2^b.3^a.c$, $c$ coprime to 6).
I claim that $f$ is continuous. Indeed, ... | 19 | https://mathoverflow.net/users/14094 | 310164 | 135,021 |
https://mathoverflow.net/questions/310168 | 2 | given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $.
consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, F\times F, \mu \times \mu)$, then $ F \times F $ is mixing too, hence ergodic, $(\mu \times \mu )(A \times A )>0$.
fixed d... | https://mathoverflow.net/users/124254 | time delay ergodic theorem | Here is a revised answer, having correctly understood the question.
Let $A\_1=A$ and $A\_0=A^c$. For a sequence $\mathbf w=w\_0,\ldots,w\_{k-1}$, set
$A\_{\mathbf w}=A\_{w\_0}\cap F^{-1}A\_{w\_1}\cap\ldots\cap F^{-(k-1)}A\_{w\_{k-1}}$. That is, $A\_{\mathbf w}$ is the set of points that is inside or outside of $A$ ... | 2 | https://mathoverflow.net/users/11054 | 310172 | 135,023 |
https://mathoverflow.net/questions/310176 | 2 | I'm currently reading Richard Garner's paper Polycategories via pseudo-distributive laws, and a central construction is the lifting of the symmetric strict monoidal category 2-monad to a pseudomonad on $Prof$, the bicategory of profunctors. I'm trying to work through all the details and they are quite messy. I was wond... | https://mathoverflow.net/users/104294 | When does a 2-functor or 2-monad of Cat lift to a psuedofunctor or pseudomonad on Prof? | Sec. 6 of [this paper](https://arxiv.org/pdf/1612.03678.pdf) should answer your question.
| 3 | https://mathoverflow.net/users/104432 | 310183 | 135,025 |
https://mathoverflow.net/questions/310184 | 4 | I have the impression, that random graphs and random matrices seem to be perceived and treated as separate areas of interest; I'm not an expert in either of the subjects, so maybe my impression is wrong.
As there is a one to one correspondence between directed graphs with self-loops and square matrices with real entr... | https://mathoverflow.net/users/31310 | What is the Essential Difference Between Random Matrices and Random Graphs? | I suppose the main point is that the typically studied random graph models are not directed or weighted and they generally don't have self loops. Under your correspondence, this means they are limited to symmetric matrices whose diagonal entries are zero and whose off-diagonal entries are either zero or one - a rather ... | 7 | https://mathoverflow.net/users/4362 | 310188 | 135,028 |
https://mathoverflow.net/questions/310178 | 7 | I am in my final year of my doctoral study in Mathematics, where my research topic is $p$-groups, specifically classification of $p$-groups by [coclass](https://en.wikipedia.org/wiki/Coclass). My work involves a great deal of computation in [GAP](https://www.gap-system.org/). I really like programming and have knowledg... | https://mathoverflow.net/users/89515 | Research in applied algebra | In the UK, there is the [Applied Algebra and Geometry Research Network](https://www.nottingham.ac.uk/Mathematics/Research/Algebra-and-Analysis/AppliedAlgebraGeometry.aspx). You could browse the list of former speakers and abstracts for ideas.
The University of St Andrews has a [strong group](http://www-maths.mcs.st-a... | 9 | https://mathoverflow.net/users/8103 | 310189 | 135,029 |
https://mathoverflow.net/questions/310198 | 5 | Let $(W,S)$ be a Coxeter system. For any subset $I\subseteq S$, we can have the *parabolic Kazhdan-Lusztig polynomial* $P\_{x,w}^I(q)$ with respect to $I$.
Now consider $I\subseteq J\subseteq S$. Both $(W,S)$, $(W\_J,J)$ are Coxeter systems.
Since $I\subseteq J$, we get the *parabolic Kazhdan-Lusztig polynomial* $\... | https://mathoverflow.net/users/110229 | Parabolic Kazhdan-Lusztig polynomial coincide? | Yes, that's true. The standard recursive constructions will give you this fact easily, because the only group elements involved in $P\_{x,w}^I$ are those which are $\leq w$ w.r.t. the Bruhat order. If $w\in W\_J$, then all those elements are themselves contained in $W\_J$.
| 5 | https://mathoverflow.net/users/3041 | 310204 | 135,034 |
https://mathoverflow.net/questions/310196 | 13 | First of all I am new to the field of embedding one manifold into another other.
I have recently come across with the paper "Embedding Riemannian manifolds by their heat kernel" by P. BERARD, G. BESSON, S. GALLOT (published in Geometric and Functional Analysis in 1984), who prove that one can embed a closed Riemannia... | https://mathoverflow.net/users/41686 | Embedding Riemannian manifolds into some infinite dimensional manifolds? | There is a more general result. Fix an even Schwartz function $\newcommand{\bR}{\mathbb{R}}$ $w:\bR\to[0,\infty)$.
Let $\Delta$ be the Laplacian of the compact connected Riemann manifold $(M,g)$, $\dim M=m$. Its eigenvalues are
$$0=\lambda\_0< \lambda\_1\leq \lambda\_2\leq \cdots$$
where each eigenvalue appears ... | 14 | https://mathoverflow.net/users/20302 | 310205 | 135,035 |
https://mathoverflow.net/questions/310161 | 5 | In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that
$$
d(A, B) \le W(\mu|\_A, \mu|\_B),
$$
where
* $d(A, B):= \inf\_{a \in A,\;b \in B}d(x,y)$ is the distance between $A$ and $B$.
* $\mu|\_A$ defined by $\mu|\_A(C):= \mu(A\cap ... | https://mathoverflow.net/users/78539 | Hausdorff distance is a lower (or upper bound) for what probability metric? | A general note is that the answer depends heavily on the properties of $\mu$.
First a note that in general $d\_H(A,B) \not \le C \cdot W\_p(\mu|\_A,\mu|\_B)$ for $p\in[1,\infty)$ and some $C>0$. Though it's true for the case $p = \infty$. Here the example: Let $\mu\_\lambda = (1-\lambda) \delta\_x + \lambda \delta\_... | 5 | https://mathoverflow.net/users/123897 | 310206 | 135,036 |
https://mathoverflow.net/questions/310165 | 8 | Given a complex simple Lie algebra $\mathfrak{g}$ of rank $n\in\mathbb{N}$ with $n$ sufficiently large (say $n\ge10$), is there a way to determine whether $\mathfrak{g}$ contains a simple subalgebra of a prescribed type with rank "close" to $n$? For example, if $\mathfrak{g}$ is of type $B\_n$ with $n\ge10$, does $\mat... | https://mathoverflow.net/users/117370 | Simple Subalgebras of Simple Lie Algebras | The answer to the question in your example is no in general: $B\_n$ does not contain $C\_{n-2}$ for large $n$. To see this, observe that $B\_n$ has an irreducible orthogonal representation $V$ of dimension $2n+1$. The Weyl dimension formula shows that for any simple Lie algebra, the dimension of an irreducible represen... | 10 | https://mathoverflow.net/users/23291 | 310209 | 135,038 |
https://mathoverflow.net/questions/310044 | 2 | Say I have a set of $(n-1)$ linearly independent vectors $\mathbf{v}\_i$ of dimension $n$ with entries $\pm1$. I am interested in finding the $n-$dimensional vector $\mathbf{u} $which is normal to the hyperplane spanned by the $\mathbf{v}\_i$. In other words, $\mathbf{u}$ is orthogonal to each of the $\mathbf{v}\_i$, w... | https://mathoverflow.net/users/94774 | fast way to calculate normal to set of vectors with $\pm$1 entries | To recap: we are given $n-1$ linearly independent vectors in ${\bf R}^n$ with $\pm 1$ entries. The original Q1 asked, in effect, if there is always a vector with $\pm 1$ entries that is orthogonal to all of the given ones. A counter examples is provided for $n=3$ by taking
$$
v\_1 = e\_1+e\_2+e\_3\quad,\quad v\_2=e\_1+... | 0 | https://mathoverflow.net/users/763 | 310216 | 135,041 |
https://mathoverflow.net/questions/309946 | 38 | *This is a crosspost from [this MSE question](https://math.stackexchange.com/q/2389376/39599) from a year ago.*
---
Finite groups are cancellable from direct products, i.e. if $F$ is a finite group and $A\times F \cong B\times F$, then $A \cong B$. A proof can be found in [this note](http://www.math.harvard.edu/~... | https://mathoverflow.net/users/21564 | Is there a non-trivial group $C$ such that $A*C \cong B*C$ implies $A \cong B$? | For $C=\mathbb{Z}/2\mathbb{Z}$, it follows from the [Kurosh subgroup theorem](https://en.wikipedia.org/wiki/Kurosh_subgroup_theorem) that $A\ast C \cong B\ast C$ implies that $A\cong B$. Denote $C\_1 \cong C\_2\cong \mathbb{Z}/2\mathbb{Z}$, and let $\varphi: A\ast C\_1 \to B\ast C\_2$ be an isomorphism.
Then $\varph... | 33 | https://mathoverflow.net/users/1345 | 310218 | 135,042 |
https://mathoverflow.net/questions/310036 | 3 | I'm confused about the precise definition of an inner automorphism of an algebraic group. Here is what Milne says in his book on algebraic groups:
Let $k$ be a field, let $\overline{k}$ be an algebraic closure, and let $G$ be an algebraic group over $k$. Let $Z$ (or $Z(G)$) denote the center of $G$. Then an automorph... | https://mathoverflow.net/users/64244 | Inner automorphisms of algebraic groups | The inner automorphisms of $G$ form an abstract group, whereas $G/Z$ is an algebraic group (i.e., group scheme of finite type over the field $k$), so you can't say that one is equal to the other --- they are different types of objects. By $(G/Z)(k)$ Milne means the group of $k$-rational points of $G/Z$, which is an abs... | 6 | https://mathoverflow.net/users/128701 | 310220 | 135,044 |
https://mathoverflow.net/questions/310224 | 5 | Let $M$ be a countable transitive model of (enough of) ZFC. Mostowski's Absoluteness Theorem says that $\Pi^1\_1$ statements are absolute between $M$ and larger models, in particular, between $M$ and the universe $V$.
For general $M$, this cannot be extended to $\Sigma^1\_2$ statements, see Andrés Caicedo's answer he... | https://mathoverflow.net/users/16107 | Buying more absoluteness for countable transitive models? | First, it is consistent that there is a ctm but no $\Sigma^1\_2$-correct ctm, for example if $V$
is the minimal model with a ctm. If there is a model of ZFC $M$
containing all the reals (e.g., if there is an inaccessible), then there is a projectively correct ctm: take the transitive collapse $H$
of a countable elem... | 8 | https://mathoverflow.net/users/102684 | 310225 | 135,047 |
https://mathoverflow.net/questions/310228 | 0 | Suppose I have the following companion matrix ($d\times d$)
[The companion matrix A](https://i.stack.imgur.com/KgkoY.jpg). $1 \geq p \geq q \geq 0$. Let $x$ ($d\times 1$) be the all one vector and my underlying problem is to analyze the first entry of $A^nx$ for some large $n$. Even if the close form doesn't exist, we ... | https://mathoverflow.net/users/128707 | Bounding/approximating the largest eigenvalue of the special case of companion matrix | First of all, we can scale your matrix so there is only one parameter, so let's say $p = 1$ for simplicity.
It looks to me like the characteristic polynomial of your matrix is
$$ P(\lambda) = \frac{\lambda^{n+1} - \lambda^n + q^n - q^{n+1}}{\lambda - q} $$
Thus $P(1) = q^n$ and $P(q) = (n+1) q^n - n q^{n-1}$. In part... | 0 | https://mathoverflow.net/users/13650 | 310230 | 135,048 |
https://mathoverflow.net/questions/304321 | 3 | Let us suppose that $X\_1,\ldots,X\_n$ with $n\ge1$ are iid random variables such that $\operatorname EX\_1=0$ and $\operatorname E|X\_1|^s<\infty$ with some $s>2$ and define the DFT of $X\_1,\ldots,X\_n$ by setting
$$
D\_n(\omega)=n^{-1/2}\sum\_{t=1}^nX\_te^{-it\omega}
$$
for $n\ge1$ and $\omega\in[-\pi,\pi]$, where $... | https://mathoverflow.net/users/46211 | Expected value of the maximum of the periodogram | Here is a sketch. Feel free to ask for clarifications if my writing gets too terse or confusing in places :-).
First recall the Bernstein (a.k.a. Hoeffding, Chernov, etc.) bound. If $Y\_m$ are mean $0$
independent random variables bounded by $s$, then for $Y=\sum\_{m=1}^n Y\_m$, we have for every
positive $t$,
$$
P... | 3 | https://mathoverflow.net/users/1131 | 310232 | 135,050 |
https://mathoverflow.net/questions/310210 | 9 | Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is finite and is called the genus of the numerical semigroup.
The sequence [A007323](https://oeis.org/A007323) in the OEIS ... | https://mathoverflow.net/users/30158 | Reference for Kakutani result on power sum bases of symmetric functions | I think Zagier must have been thinking of the following papers of Kakeya, instead of Kakutani
>
> Kakeya, S.: On fundamental systems of symmetric functions. [I](https://doi.org/10.4099/jjm1924.2.0_69), [II](https://doi.org/10.4099/jjm1924.4.0_77). [Jap. J. Math.**2**, 69–80 (1925)](https://doi.org/10.4099/jjm1924.2... | 8 | https://mathoverflow.net/users/2384 | 310234 | 135,051 |
https://mathoverflow.net/questions/310203 | 2 | Many definitions of $2$-categories are given as categories equipped with some extra structure encoded by some functors and some natural (or extranatural) transformations between these functors (or with $Id$ functors, etc.), satisfying some commutativity conditions between them.
Sometimes, one of those "equipments" ar... | https://mathoverflow.net/users/95265 | Relaxing a natural isomorphism to a natural transformation to obtain a more general $2$-category | One situation in which this can be done if the original structure can be described as a pseudo-algebra structure for some 2-monad. In this case, to make the constraints noninvertible one can consider instead lax algebras or colax algebras for the same 2-monad. However, this process tends to produce only [unbiased](http... | 3 | https://mathoverflow.net/users/49 | 310240 | 135,052 |
https://mathoverflow.net/questions/310119 | 8 | Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)?
Recall from SGA 1 that the theory of Galois categories is developed to construct the étale fundamental group of a connec... | https://mathoverflow.net/users/5337 | Galois categories for topological spaces? | The answer is yes (with mild hypothesis on the space). Moreover the topological situation is simpler, and this was very likely Grothendieck's inspiration.
To see this you need two facts.
First taken from Szamuley's book [Galois Groups and Fundamental Groups](http://www.cambridge.org/catalogue/catalogue.asp?isbn=978... | 7 | https://mathoverflow.net/users/11682 | 310243 | 135,054 |
https://mathoverflow.net/questions/310110 | 4 | Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta\_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{c},\mathbf{x} \rangle$ is the inner product between $\mathbf{c}$ and $\mathbf{x}$.
**Question**: Given $\mathbf{c}$ ... | https://mathoverflow.net/users/115803 | Fast projection onto a subspace | As noted in the comments, this problem is not really a research level problem. Afaik, versions of it were originally solved in the 50s.
Here is an entire survey that discusses efficient algorithms (including linear-time procedures) for this problem as well as generalizations of it: M. Patriksson, *[A survey of classi... | 2 | https://mathoverflow.net/users/8430 | 310250 | 135,056 |
https://mathoverflow.net/questions/309007 | 3 | Suppose $K\_3$ is the Kronecker quiver with 3 arrows, and $M^{ss}\_{(2,2)}(K\_3,(-1,1))$ is the moduli space of semi stable representation of dimension $(2,2)$ wrt the weight $(-1,1)$. It is claim in the introduction of
<https://arxiv.org/pdf/math/0010251.pdf>
that this moduli space is isomorphic to $M\_{\mathbb{P}^2}... | https://mathoverflow.net/users/48616 | $M^{ss}_{(2,2)}(K_3,(-1,1))$ is isomorphic to $M_{\mathbb{P}^2}(0,2)$ | The construction of the equivalence is described in the articles the author cites, but these are somewhat old references so it might be helpful if I try to translate it into modern language.
It is known that $\mathrm{Coh}(\mathbb{P}^2)$ is derived equivalent to the category $\mathrm{mod}\Lambda$, where $\Lambda$ is a... | 2 | https://mathoverflow.net/users/nan | 310257 | 135,060 |
https://mathoverflow.net/questions/310227 | 8 | Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a\_p(f)$ is a $p$-adic unit (to make sense of this in general, one needs to choose a prime ideal above $p$ in the field $K\_f$ of Fourier coefficie... | https://mathoverflow.net/users/6506 | Existence of newforms which are non-ordinary at a given prime | Given p and k, it's clear we can find a CM-type newform of weight k and some level which is supersingular at p (just choose an imaginary quadratic field in which p is inert, and some sufficiently large conductor away from p).
So it suffices to find a second newform that is congruent to the first one mod p and is *no... | 6 | https://mathoverflow.net/users/2481 | 310262 | 135,061 |
https://mathoverflow.net/questions/310245 | 3 | I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?
This seems to be a well-known result, but I can not find it in standard references such as Stein's Harmonic Analysis and Grafakos's classical Fourier Analysis.
Than... | https://mathoverflow.net/users/35702 | Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$ | As mentioned in the comments, the Fourier transform $\hat f$ of a function in $L^{\frac{2n}{n+1}}({\bf R}^n)$ is *a priori* only defined as an element of the dual space $L^{\frac{2n}{n-1}}({\bf R}^n)$ (as per the Hausdorff-Young inequality), and so cannot immediately be restricted to the measure zero set $S^{n-1}$ unle... | 7 | https://mathoverflow.net/users/766 | 310264 | 135,063 |
https://mathoverflow.net/questions/310244 | 2 | Few days I asked this question (<https://math.stackexchange.com/questions/2907733/simple-ordinal-question>) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta\_x$ (where $x<\omega\_1$) and asked about its relation to the function $x \mapsto \omega^{CK}\_x$. And it seems ... | https://mathoverflow.net/users/112385 | Formal definition of this ordinal? | This is actually much simpler than you may suspect: $\omega\_\alpha^{CK}$ is well-defined for **every** ordinal $\alpha$, not just the countable ones, if we use the set-theoretic as opposed to computability-theoretic definition. Specifically, we define $\omega\_\alpha^{CK}$ as the unique ordinal $\eta$ such that
* $L... | 6 | https://mathoverflow.net/users/8133 | 310269 | 135,066 |
https://mathoverflow.net/questions/310270 | 14 | Recall that the Stiefel-Whitney classes of a smooth manifold are defined to be those of its tangent bundle - this definition doesn't extend to topological manifolds as they don't have a tangent bundle. Wu's theorem states that for a closed smooth manifold, $w = \operatorname{Sq}(\nu)$. The expression $\operatorname{Sq}... | https://mathoverflow.net/users/21564 | Is the top Stiefel-Whitney number of a topological manifold the Euler characteristic mod two? | As you say, we define $w\_n = \sum \text{Sq}^i \nu\_{n - i}$, where $\nu\_{n-i}$ is the Wu class, the class such that $\nu\_{n-i} \cup c = \text{Sq}^{n-i} c$ for $c \in H^{i}$. So as a corollary we have $\text{Sq}^i \nu\_{n - i} = \nu\_i \cup \nu\_{n-i}$.
Because $\nu\_j$ vanishes for $j > n/2$, the sum over $i$ is ... | 14 | https://mathoverflow.net/users/40804 | 310275 | 135,069 |
https://mathoverflow.net/questions/310271 | 5 | We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov homology or HOMFYLPT homology from there?
When googling, one can find a lot of papers containing certain categorifications of ... | https://mathoverflow.net/users/101335 | Categorifying skein algebras? | Much of the research in knot homology has been about categorifying these algebras!
Khovanov's paper [math/0103190](http://de.arxiv.org/pdf/math/0103190.pdf) is devoted to defining and studying the Temperley-Lieb 2-category which is a categorification of the Temperley-Lieb category. (The TL algebras arise and endomor... | 2 | https://mathoverflow.net/users/438 | 310281 | 135,072 |
https://mathoverflow.net/questions/310268 | 0 | Given a random $d$-regular graph on $n$ nodes, what is the expected number of common neighbors between two nodes?
I don't know if it is as simple as just assuming that each neighbor of the first node has a $\frac{d}{n}$ probability of being a neighbor of the second, as the set of $d$-regular graphs on $n$ nodes is di... | https://mathoverflow.net/users/128739 | The expected value of common neighbors on a random regular graph | Yep, it is $\frac{d(d-1)}{n-1}$ in general. Fix a $d$-regular graph and average over the action of the permutation group on the vertices. We are interested in the event $S(x,y,z)=$"$x$ is connected to $y$ and $z$". If $E$ is the expected number of neighbors, then $\sum\_{x,y,z}P(S(x,y,z))=n(n-1)E$, summing over $x$ fir... | 2 | https://mathoverflow.net/users/1131 | 310282 | 135,073 |
https://mathoverflow.net/questions/310280 | 10 | Let $ \mathcal C $ be a monoidal category. Then $ \mathcal C $ is both a left and right module category over itself. Moreover, the Drinfeld centre of $ \mathcal C $ can be defined as the category of functors from $ \mathcal C $ to itself which commute with these module category structures.
$$
Z(\mathcal C) = Fun\_{\mat... | https://mathoverflow.net/users/438 | Generalization of Drinfeld double to comodule algebras | Such an algebra exists. As far as I am aware, the algebra was first described in chapter 6 of [The blob complex](https://arxiv.org/abs/1009.5025) by Morrison and Walker. In this paper, the algebra is construct from a diagrammatic calculus for the module and tensor category rather than from $H$ and $A$ directly.
Since... | 6 | https://mathoverflow.net/users/4002 | 310286 | 135,075 |
https://mathoverflow.net/questions/310278 | 2 | Suppose I collect $2n$ independent samples of a probability density function $f$, which are separated into pairs $\{X\_i^1, X\_i^2\}$ for $1\leq i\leq n$. Suppose I now consider the set of all $2^n$ sequences obtained by taking one sample from each pair. Are there any existing results comparable to the law of large num... | https://mathoverflow.net/users/125803 | A modest generalization of the law of large numbers | In the case where the $X$'s take finitely many values, you can prove what I claimed using the max-flow min cut theorem. I have not written it down, but this should be extendable to the general case.
Here's what I mean.
Suppose you have two finite sets $A$ and $B$, equipped with probability measures $p$ and $q$. ($A... | 2 | https://mathoverflow.net/users/11054 | 310288 | 135,076 |
https://mathoverflow.net/questions/310247 | 5 | I asked the question on MSE.
<https://math.stackexchange.com/questions/2898377/what-is-the-stone-space-of-the-free-sigma-algebra-on-countably-many-generators>
The answer I got, however, seems disputed. I just thought that someone here could answer the question for sure. Many thanks.
| https://mathoverflow.net/users/126821 | What is to Stone space of the free sigma-algebra on countably many generators? | You got a wrong answer on Math Stackexchange from Daron. The free Boolean algebra on countably many generators is the Boolean algebra of clopens of $2^\omega$ (topologized with the product topology), and the free $\sigma$-algebra on countably many generators is the Baire $\sigma$-algebra of $2^\omega$, which, as $2^\om... | 5 | https://mathoverflow.net/users/61785 | 310292 | 135,078 |
https://mathoverflow.net/questions/310285 | 10 | In study of the cohomology ring of the Grassmannians, which is usually known as **Schubert calculus**, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma\_\lambda\}$. One of the most properties of this basis is **positivity**, the fact that for any two basis elements $\sigma\_\lambda$ an... | https://mathoverflow.net/users/126606 | Proving Positivity for Schubert Calculus | I would say there are three basic reasons for / proofs of positivity.
1. *Geometry.* [Kleiman 1973] proves that the number one's trying to compute is the number of points in a *transverse* intersection of cycles. Ergo, a nonnegative number.
2. *Combinatorics.* Present the cohomology ring of $Gr(k,n)$ as a quotient of... | 18 | https://mathoverflow.net/users/391 | 310295 | 135,079 |
https://mathoverflow.net/questions/310290 | 10 | Let $\Gamma$ be an arithmetic lattice in a linear algebraic $\mathbb{Q}$-group $\mathbf{G}$, that is, $\Gamma$ is a subgroup of $\mathbf{G}(\mathbb{Q})$ that is commensurable with $\mathbf{G}(\mathbb{Z})$.
For a prime $p$, we can consider $\Gamma$ as a subspace of $\mathbf{G}(\mathbb{Q}\_p)$. My question is:
What d... | https://mathoverflow.net/users/20140 | What does the $p$-adic closure of an arithmetic lattice look like? | Suppose $G$ is $\mathbb Q$ simple (i.e. has no connected normal algebraic subgroups which are defined over $\mathbb Q$) and is simply connected (i.e. $G(\mathbb C)$ is simply connected). Assume also that $G(\mathbb R)$ is not compact. With these assumptions, the closure of an arithmetic lattice in $G({\mathbb Z}\_p)$ i... | 8 | https://mathoverflow.net/users/23291 | 310298 | 135,080 |
https://mathoverflow.net/questions/310291 | 27 | I've recently heard about an idea of Serre that for each finite group $G$ there exists a group scheme $X$ such that for each field $K$ the group $X(K)$ is naturally isomorphic to the unit group of $K[G]$. Unfortunately, the article where this fact was mentioned gave no reference, so I ask you if you know how to constru... | https://mathoverflow.net/users/88385 | Serre's remark on group algebras and related questions | It's fairly easy to do this for finite groups. In fact, the functor $R \mapsto R[G]$ is naturally representable by a ring scheme: the underlying set functor is represented by $\mathbb A^n$ where $n = |G|$, and the ring structure comes from the functor of points $R \mapsto R[G]$. Write $Y$ for this ring scheme (say over... | 24 | https://mathoverflow.net/users/82179 | 310302 | 135,081 |
https://mathoverflow.net/questions/310287 | 3 | Denote by $\Omega^2({S}^2)$ the space of DOTTED maps from the $2-$sphere $S^2$ onto itself. And consider its FREE loop space $X=\mathcal{L}(\Omega^2({S}^2))=Maps(S^1, \Omega^2({S}^2))$. I think that $\pi\_0(X)$ is $\pi\_3(S^2)\oplus\pi\_2(S^2)$ where the integer in $\pi\_2(S^2)$ measures the degree of the map $\{\cdot\... | https://mathoverflow.net/users/128744 | How to compute $\pi_0$ of $Maps(S^1, \Omega^2({S}^2, p))$ | I will assume that "dotted" means the same as "basepoint-preserving".
There are homeomorphisms $$\mathcal L\Omega^2 S^2\cong \mbox{map}\_\*(S^1\_+\wedge S^2, S^2)\cong \mbox{map}\_\*(S^3/S^1, S^2).$$ Note that there is a homotopy equivalence $S^1\_+\wedge S^2\simeq S^3\vee S^2$. Therefore, there is a homotopy equiva... | 6 | https://mathoverflow.net/users/6668 | 310305 | 135,082 |
https://mathoverflow.net/questions/310300 | 3 |
>
> Let $S$ be a connected scheme, let $\pi : \mathbb{P}\_{S}^{r} \to S$ be projective $r$-space over $S$, and let $\mathcal{E}$ be a flat and locally finitely presented $\mathcal{O}\_{\mathbb{P}\_{S}^{r}}$-module. Is $\pi\_{\ast}(\mathcal{E}(n))$ (nonzero and) flat and locally finitely presented for $n \gg 0$?
>
> ... | https://mathoverflow.net/users/15505 | Non-noetherian cohomology and base change | The answer to the question about flatness is no and here is a counterexample. Let $S\_n = \mathbf{A}^1$ with coordinate $t$ for $n\geq 1$ (over some base field). Let
$$
\mathcal{A}\_n = \mathcal{O}\_{\mathbf{P}^1}(-n)\oplus \mathcal{O}\_{\mathbf{P}^1}(n).
$$
We have a non-split extension
$$ 0\to \mathcal{O}\_{\mathbf... | 4 | https://mathoverflow.net/users/3847 | 310313 | 135,084 |
https://mathoverflow.net/questions/310316 | 15 | (Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the example of $SO(2)$. All $SO(2)$ matrices $M$ can be written as ($\theta\in [0,2\pi[$)
$$
M=\begin{pmatrix}\cos\theta & \sin... | https://mathoverflow.net/users/125359 | Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices? | **Elementary proof**. The linear space $E$ spanned by $SO\_n$ is the orthogonal of those matrices $M$ such that $\langle M,Q\rangle:={\rm Tr}(MQ)=0$ for every $Q\in SO\_n$. Let $M=SR$ be a polar decomposition, where $S\in Sym\_n^+$ and $R\in O\_n$. This decomposition is unique with $S\in SPD\_n$ if $M$ is non-singular,... | 14 | https://mathoverflow.net/users/8799 | 310321 | 135,088 |
https://mathoverflow.net/questions/310322 | 3 | Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?
| https://mathoverflow.net/users/128762 | Intersecting geodesics on a surface from non-intersecting geodesics | No. The "geodesic parametrization" of a curve is unique (up to pre-composition with a rotation) in the curve's homotopy class. You can find more information in the book "A primer on mapping class groups".
Edit: and Lee beat me to the answer by just two minutes! I'll leave this here for the reference.
| 2 | https://mathoverflow.net/users/1650 | 310333 | 135,091 |
https://mathoverflow.net/questions/310339 | 5 | Let $X$ be a connected $CW$-complex, such $\pi\_1(X)$ is torsion-free and $H\_k(X,\mathbb Z) = 0$ for all $k \geq N$ and some $N \in \mathbb N$. Then
$(1)$ Does it follow that $X$ is homotopy-equivalent to its $N$-skeleton, i.e $X \simeq X^{(N)}$ ?
$(2)$ If $(1)$ is false, does it follow that $X \simeq X^{(k)}$ fo... | https://mathoverflow.net/users/78554 | On spaces with finite homological dimension | You can take $X=BG$ where $G$ is a torsion-free, acyclic group of infinite cohomological dimension. Acyclic means $H\_k(BG;\mathbb{Z})=0$ for $k>0$, and infinite cohomological dimension implies infinite geometric dimension, so $BG$ is not homotopy equivalent to any finite dimensional CW complex $Y$. Such a group theref... | 8 | https://mathoverflow.net/users/8103 | 310350 | 135,097 |
https://mathoverflow.net/questions/310326 | 2 | Recall that there are $$\frac{n!}{\prod^n\_{i = 1}i^{k\_i}k\_i!}$$ permutations in $S\_n$ which have cycle structure $(k\_1, \dots, k\_n)$, that is to say they have exactly $k\_1$ 1-cycles, $k\_2$ 2-cycles, ... and $k\_n$ n-cycles. The cycle index of $S\_m \times S\_n$ acting on the set $\{1, \dots, n\} \times \{1, \do... | https://mathoverflow.net/users/128120 | Cycle index of $(S_n \times S_n) \rtimes C_2$ acting on matrix indices by row/column permutation and transposition | The question is about cycle types of elements of the wreath product $G = (S\_n \times S\_n) \ltimes C\_2 \cong S\_n \wr C\_2$ in its product action on $\{1,\ldots, n\} \times \{1,\ldots, n\}$.
The permutation $(\sigma, \rho, c) \in G$ is conjugate, by $(\rho^{-1},\mathrm{id}\_{S\_n},1)$, to $g = (\rho\sigma, \mathrm... | 2 | https://mathoverflow.net/users/7709 | 310351 | 135,098 |
https://mathoverflow.net/questions/310143 | 4 | Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ is an equivalence of homotopy categories and it is well defined. We define the category $\pi\_{0}M$ as the categor... | https://mathoverflow.net/users/128371 | Simplicial model categories and simplicial equivalence | Since there is an answer to the question, I think I should write it down.
There is a simple counterexample to my question: Let $M=N=sSet$ the standard model category of simplicial sets. Let $ex^{\infty}:sSet\rightarrow sSet$ the fibrant replacement functor. It is simplicial as it was noticed in the comments. $Ho(F)$... | 2 | https://mathoverflow.net/users/128371 | 310356 | 135,102 |
https://mathoverflow.net/questions/310335 | 9 | It is of course completely standard that closed orientable surfaces have even Euler characteristic. What is the most elementary proof of this?
More specifically, suppose I have a finite simplicial complex $K$ with vertices $V$, edges $E$ and faces $F$. I suppose that each edge is contained in precisely two faces, and... | https://mathoverflow.net/users/10366 | Closed orientable surfaces have even Euler characteristic | The set of vertices can be split as the union $V=V\_{\text{odd}}\cup V\_{\text{even}}$ of vertices with odd and even degrees, respectively. Since the sum of degrees over all vertices is the same as twice the number of edges, we know that $|V\_{\text{odd}}|= 0\pmod 2$. Therefore we want to establish that $|V\_{\text{eve... | 15 | https://mathoverflow.net/users/2384 | 310362 | 135,105 |
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