Newest Questions
159,019 questions
3
votes
1
answer
285
views
Examples of Lie groups where $G\to G/H$ splits topologically but not as groups
Let $G$ be a connected Lie group and $H$ a normal connected subgroup. I'm looking for examples where $G\to G/H$ has a section as topological spaces, but not as Lie groups.
My motivation here is to ...
- 740
4
votes
0
answers
106
views
Graphs and symmetries
Let us define a symmetric cut of the graph as a cut which cuts the graph into two isomorphic graphs and that the cut preserves the isomorphism i.e. the edges that are cut only connect vertices that ...
- 41
5
votes
1
answer
304
views
Recover unknown vector through shifted argmax queries
$\DeclareMathOperator*{\argmax}{arg\,max}$
I am interested in finding an efficient algorithm for the following problem:
Let $x \in [0,1]^n$ be some vector, with $x_n = 1$.
We want to recover $x$, ...
- 167
4
votes
1
answer
211
views
Erdős–Rényi random graphs — reproducing 2 inequalities
In Erdős and Renyi's 1959 paper On random graphs I
, I'm trying to reproduce, starting from Eq.\eqref{1} in their paper, the two inequalities that appear in Eq.\eqref{2}.
Eq.\eqref{1} is:
$$
P \le \...
- 43
2
votes
0
answers
132
views
For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$
Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
- 411
1
vote
0
answers
36
views
Construction of a regulariser for the boundary integral operator $\lambda\mathrm{Id} - K'$
$\newcommand\Id{\mathrm{Id}}$Assumptions and Notations :
$\Omega$ is a bounded Lipschitz domain in $\mathbb R^2$, $\Gamma$ denotes its boundary and $n$ is the normal vector to the boundary $\Gamma$,
...
- 63
5
votes
1
answer
361
views
Conductor of determinant of a 2-dimensional Galois representation divides conductor of representation
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)...
1
vote
2
answers
390
views
Solving a recurrence relation for the prime counting function?
I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ A307977(n).
I am trying to solve the following recurrence relation for the prime counting function:
$$\forall n \ge 3: \pi(...
- 3,074
3
votes
1
answer
495
views
Legendre's conjecture for semiprimes
Legendre's conjecture states that for every integer $n\ge1$, there is a prime between $n^2$ and $(n+1)^2$. This is an important problem about prime numbers that has not been solved for a long time, ...
- 61
2
votes
0
answers
116
views
Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives
If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$?
It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
- 469
9
votes
2
answers
773
views
Pullbacks of classifying spaces
In what follows all the groups will be discrete, not necessarly finite.
Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am ...
- 641
4
votes
1
answer
201
views
Does $N \mathbin{\bar{\otimes}} N^{\mathrm{op}}$ act on $L^2(N)$?
Let $N$ be a von Neumann algebra and $N^{\mathrm{op}}$ its opposite. The standard form $L^2(N)$ is an $N$-$N$-bimodule, or equivalently a module over $N \otimes_{\mathrm{alg}} N^{\mathrm{op}}$.
Does ...
- 6,406
1
vote
0
answers
126
views
$p'$-automorphisms of pro-$p$ groups
Let $p$ be a prime and $G$ be a finitely generated pro-$p$ group admitting a continuous automorphism $\phi$ of finite order relatively prime to $p$. Let $\Phi(G)$ denote the Frattini subgroup of $G$. ...
- 667
2
votes
0
answers
113
views
Selmer groups associated with Drinfeld modules
Given an elliptic curve $E_{/\mathbb{Q}}$ (or more generally, a number field) and a prime $p$, there is a standard short exact sequence
$$0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}_p/\mathbb{Z}_p\...
- 685
4
votes
0
answers
131
views
Is there a fast way to do this tensor power/trace operation?
Well, I asked this question on Math SE, and didn't get any responses, so I'm trying it here.
Given an $M*N*P$ tensor $T$, is there a fast way of computing the following "eighth-power trace"?
...
- 525
12
votes
1
answer
449
views
Can Friedman's property fail at or above a supercompact cardinal?
If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\...
- 435
0
votes
0
answers
164
views
Inf-convolution of norm 1 and norm 2 square
The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is
$$
h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) .
$$
We can prove that if $f,g$ are convex functions, then $h$ is convex.
...
- 129
5
votes
0
answers
377
views
Topologies on diffeomorphisms groups
Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual ...
- 426
1
vote
0
answers
141
views
Can a closed null-homotopic curve be filled in by a disc?
Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...
- 13.6k
1
vote
2
answers
224
views
Is there always at least one Markov number between $n$ and $n^2$, for $n > 2$?
I'm new to this site, so maybe it's a bad question, but I've been researching Markov numbers for a few days and I haven't found much research about them other than Wikipedia and a few sites. What I'm ...
- 61
2
votes
1
answer
237
views
Small deviations of real log-concave random variable
I am working with a log-concave real random variable, that has a density $f(x) = \exp(-\varphi(x))$ with $\varphi$ convex. Assuming that $X$ is centered and has unit variance ($\mathbb{E}X=0$, $\...
- 23
4
votes
1
answer
181
views
Is every area-minimizing cone a level set of a least-gradient function?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons ...
- 5,048
1
vote
0
answers
193
views
On Lubkin's works on p-adic cohomology
I am curious to know what the things are going after Lubkin's several works on p-adic cohomology. As Math.Sci.Net shows, his papers on this topic have a few citations despite of the titles which ...
- 31
13
votes
1
answer
385
views
Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?
There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...
- 56.9k
6
votes
0
answers
511
views
Extensions of p-adic number fields
Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...
- 71
1
vote
1
answer
341
views
Form of minimax estimator
Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\...
- 6,541
2
votes
0
answers
169
views
How a circle $S^1$ acts on the Cayley plane $OP^2$ with exactly three fixed points?
The complex projective $CP^2$, the quaternionic projective space $HP^2$, and the octonionic projective space $OP^2$ each admit a circle action with $3$ fixed
points.
The circle action on $HP^2$ can be ...
- 103
5
votes
0
answers
322
views
Approximating $\zeta^{(r)}(s)$ by a sum
Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
- 20.2k
4
votes
1
answer
237
views
Spectral density of symmetrized Haar matrix
Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can'...
- 1,608
2
votes
0
answers
45
views
Torelli theorem for veronese double cone(reference needed)
Let $Y$ be a smooth Veronese double cone, which is a smooth del Pezzo threefold of degree one, which can be regarded as a weighted hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. I was wondering ...
- 1,982
0
votes
1
answer
127
views
A sufficient condition for a collection of open sets of a manifold to contain all open sets
Question
Let $k\geq 0$ be an integer and let $M$ be a topological $n$-manifold. Let $\mathcal{U}$ be a set of open sets of $M$ which satisfies the following closure properties:
(1). Let $U\subset M$ ...
- 2,292
2
votes
0
answers
113
views
From rational to integral generators of Néron–Severi group
Suppose I've found rational generators for the Néron–Severi group $\mathrm{NS}(A)$ for an abelian
variety over $\mathbb{C}$. How would I check if they are integral generators for $\mathrm{NS}(A)$. Are ...
- 91
2
votes
1
answer
219
views
Homomorphism between ideal sheaves of codimension $2$?
Let $X$ be a smooth projective variety with two different reduced subschemes $Z,Y\subset X$ which induced from ideal sheaves $\mathscr{I}_Z,\mathscr{I}_Y$. If codimensions of them are both larger than ...
- 249
3
votes
1
answer
373
views
reference for reading Schoen Yau positive mass theorem proof II
I am trying to read the paper by Schoen and Yau, Proof of the Positive Mass Theorem II. The notation is very different from what I am familiar with (basically Robert Wald's book on general relativity)....
- 61
2
votes
0
answers
104
views
Unordered configuration space with non-distinct points
Consider a topological space $X$, a natural number $n>0$ and
the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if
there is a permutation $\sigma$ ...
- 1,035
2
votes
1
answer
285
views
Deligne-Lustzig varieties locally closed schemes
I have a couple of questions about basic properties of of Deligne-Lustzig varieties introduced in the seminal paper "Representations of Reductive Groups Over Finite Fields" [DL76].
The ...
- 6,038
8
votes
1
answer
264
views
Does the continuous image of a disc contain an embedded disc?
Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
- 13.6k
1
vote
0
answers
146
views
Is the functor $\mathrm{Hom}(\mathrm{spec}\,k[x^{1/{p^\infty}}]/(x), -)$ from the category of finite commutative group schemes exact?
Question. Let $B \twoheadrightarrow C$ be a fully faithful homomorphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. ...
- 199
22
votes
3
answers
3k
views
Is the sum of the reciprocals of the products of pairs of coprime positive integers and their sums equal to 2?
Does the following hold?:
$$
\sum_{a, b \in \mathbb{N}^+, \ \gcd(a,b) = 1} \frac{1}{ab(a+b)} \ = \ 2
$$
Numerical computations suggest this may hold, but on the other hand
it would be quite ...
- 19.6k
5
votes
0
answers
215
views
Bound the volume of a variety of general type by its topology
Given any sequence of $n$-dimensional complex projective manifolds $\{X_i\}_{i=1}^{\infty}$ such that $X_i$ are diffeomorphic to each other and have ample canonical divisors $K_{X_i}$. Are the ...
- 91
7
votes
0
answers
1k
views
What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
- 157
2
votes
0
answers
97
views
Lower & upper bound on the maximal component given the system of power sums
Given a non-negative vector $x=(x_1,x_2,\dots,x_n)\in\mathbb{R_{>0}^n}$ and $m\in\mathbb{N}$, construct a system of power sum symmetric polynomials (or norms, if you like)
$$
\begin{cases}
x_1+x_2+\...
- 21
16
votes
2
answers
1k
views
The Stable Set Conjecture
A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation
$$n\in \mathcal S \iff dn\in \mathcal S$$
holds for almost all positive integers $n$. ...
- 791
11
votes
0
answers
3k
views
Eric T. Sawyer's proof of Fourier restriction conjecture
Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
- 111
3
votes
1
answer
161
views
Approximating continuous functions from $K\times L$ into $[0,1]$
Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
- 5,529
0
votes
1
answer
159
views
Can any Clifford module bundle be extended to a Dirac bundle?
I assume that the question in the title is clear, so let me talk about its relevance:
According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem
\begin{equation}\tag{1}
\mathrm{ind}...
- 339
4
votes
0
answers
313
views
What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
- 571
3
votes
1
answer
221
views
Asymptotics for number of $p$-regular partitions of $n$
The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
- 571
4
votes
0
answers
291
views
Has anyone studied the derived category of Higgs sheaves?
Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-...
- 9,019
-1
votes
1
answer
132
views
What is an "open Baire set"?
In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
- 297