All Questions
21 questions from the last 7 days
6
votes
3
answers
359
views
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
- 128k
8
votes
1
answer
325
views
Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 501
2
votes
1
answer
189
views
Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 415
4
votes
0
answers
149
views
Unramified Galois cohomology
Let $k$ be a local field with absolute Galois group $\Gamma_k$, inertia subgroup $I_k \subset \Gamma_k$, and residue field $\mathbb{F}$. Let $M$ be a finite Galois module over $k$.
The unramified ...
- 21.4k
3
votes
0
answers
131
views
On the convergence of $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$
Let $\alpha$ be an irrational number, and $\left\{ \cdot \right\}$ denotes the fractional part function. We have focused on how $\left\{ n! \alpha \right\}$ distributes in this MO question. And now I ...
5
votes
0
answers
106
views
Derived functors and functorial resolutions/(co)fibrant replacements
I will begin with some context; the question itself is highlighted below. This is all for some notes I am writing personally on homological algebra, amongst other things.
To construct derived functors,...
3
votes
1
answer
76
views
Full asymptotics near 1 of the generating function for integer partitions
Consider the generating function for integer partitions
$$
P(q)=\frac{1}{\prod_{k=1}^{\infty}(1-q^k)}=\sum_{n=0}^{\infty}
p(n)q^n\ ,
$$
where $p(n)$ is the number of integer partitions of $n$.
I am ...
- 23.1k
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
0
votes
1
answer
61
views
A combinatorial linear programming problem
$\newcommand\S{\mathscr S}$Let $\S$ be a collection of nonempty subsets of a finite set $S$ such that $A\not\subset B$ for any distinct $A$ and $B$ in $\S$.
Does then there always exist a function $f\...
- 128k
1
vote
0
answers
63
views
Reference request: Proof theory in $W_1^1$
Buss defined $V_2^1$ as a second-order bounded arithmetic corresponding to $\mathsf{PSPACE}$.
Later, Skelley introduced $W_1^1$, a third-order bounded arithmetic of $\mathsf{PSPACE}$.
Since the ...
- 11
3
votes
0
answers
82
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
3
votes
0
answers
40
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
1
vote
0
answers
32
views
Does it suffice to consider the limit only along the power paths?
$\newcommand\R{\Bbb R}$For $a=(a_1,\dots,a_n)\in\R^n$, let
$$L(a):=\lim_{x\downarrow0}R_a(x)$$
if the limit exists, where
$$R_a(x):=\frac{x^a}{F(x)},$$
$x=(x_1,\dots,x_n)\in\R^n$, $x\downarrow0$ means ...
- 128k
1
vote
0
answers
57
views
Martingale decomposition in Aldous' famous 1997 paper
I was reading Aldous' famous 1997 paper (Aldous, David. "Brownian excursions, critical random graphs and the multiplicative coalescent." The Annals of Probability (1997): 812-854.)
In his ...
- 111
1
vote
0
answers
34
views
Vertex coloring of the Rado graph
Is there a reference for the following fact about the Rado graph (the random countable graph) which came up in an answer to this question?
If the vertices of the Rado graph $G=(V,E)$ are colored with ...
- 13.4k
0
votes
0
answers
36
views
Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
1
vote
0
answers
41
views
Can conditional distributions with respect to a sufficient sub-$\sigma$-algebra be represented by a single Markov kernel?
Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathcal{P}$ be a collection of probability measures on this space. A sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is said to be ...
- 111
1
vote
0
answers
14
views
Literature on Fréchet quasi-coherent sheaves
I've heard/read it said many times that "the good notion of quasi-coherent sheaf for complex manifolds is that of a Fréchet quasi-coherent sheaf", and the standard citation to which I've ...
- 1,139
0
votes
0
answers
25
views
When localization commutes with arbitrary intersection of ideals
For commutative ring with identity we know that in general localization dose not commute with arbitrary intersection of ideals. I am looking for a paper that consider equivalent condition for rings ...
0
votes
0
answers
13
views
Unitaries that setwise fix an algebra under conjugation
Let $M_d(\mathbb{C})$ denote the algebra of $d \times d$ complex matrices. Consider the algebra
$$\mathcal{A} = \bigoplus_{i=1}^r I_{d_i} \otimes M_{d_i}(\mathbb{C})$$
for some choice of $d_1, \ldots, ...
- 2,339
0
votes
0
answers
25
views
Can I get a long exact homology sequence from a short exact sequence of graded commutative differential algebras over a field?
The category of differential graded commutative algebras over a field of characteristic 2 seems to have some not so nice properties. I could not find an answer in StackExchange as to why this category ...
- 35