Trending questions
159,052 questions
1
vote
1
answer
91
views
Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
0
votes
1
answer
169
views
Existence of a "universal" measure-preserving transformation on the unit interval
Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
user545331
1
vote
0
answers
81
views
Is every homogeneous line bundle pulled back from the quotient stack?
Let $G= \mathbb{G}_m^k$ act on a variety $X$.
Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$.
Does it ...
- 323
3
votes
1
answer
158
views
Three congruences for a Perrin-like sequence and pseudoprimes
Let $ V(n) $ be defined by the recurrence relation:
$$
V(n) = 3\,V(n-2) + V(n-3)
$$
with the initial conditions:
$$
V(0) = 3, \quad V(1) = 0, \quad \text{and} \quad V(2) = 6.
$$
If $ n $ is an odd ...
1
vote
0
answers
149
views
Inequalities in the classic proof of perfect matching in Erdős–Rényi graph
I am checking the classic paper by Erdős and Rényi, "On the existence of a factor of degree one of a connected random graph" with the link here. I am curious about the computation of the ...
- 97
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
2
votes
0
answers
70
views
Is the hypothesis "uniformly equivalent" needed?
I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows:
Assume $\mathscr{H}$ is a Hilbert space of ...
- 151
1
vote
2
answers
268
views
Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?
Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that
The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which ...
- 515
8
votes
1
answer
199
views
Topological property of the space of probability measures
Suppose that $\mathbb{P}$ is the metric space of Borel probability measures on the interval $[0,1]$ equipped with the topology of $w^*$ convergence.
Consider also $\mathbb{P}_{ac}, \mathbb{P}_{s}$ the ...
1
vote
0
answers
58
views
'Invert' perturbed vorticity equation to forced Euler system
Given the vorticity form of the Euler equations in $2D$ with stream function $\psi$
\begin{align}
\omega_t + \nabla^\perp \psi \cdot \nabla\omega &= 0 \\
\Delta \psi = \omega
\end{align}
we know ...
- 255
0
votes
0
answers
37
views
Maximise norm over the boundary of a convex set
Let $K\subset \mathbb R^2$ be compact, convex and connected. What is the know numerical scheme to find the extremal points of $K$?
Denote by $\partial K$ the collection of all extremal points of $K$. ...
- 1,399
1
vote
1
answer
52
views
Comparability of elements in a Latin square based on a few rows
Let $\Pi=\{\pi_1,\pi_2,\dots,\pi_n\}$ be the rows of an $n\times n$ Latin square (the order of the rows does not matter).
Each row $\pi_i$ induces an order $\prec_i$ on the elements of $[1,n]$, where $...
- 235
4
votes
1
answer
162
views
Finite group is Dedekind iff for every irrep, every element acts as identity or has all eigenvalues $\ne 1$
Consider the following claim: a finite group $G$ is Dedekind $\iff$ for every irrep $\rho$, and every $g \in G$, $\rho(g)$ either is identity matrix or has all eigenvalues $\ne 1$.
Is this claim true?
...
- 393
1
vote
1
answer
41
views
Lower spectral radius of matrices with an invariant subspace
Let $\mathcal{A} = \{ A_1, \ldots, A_J, : A_j \in \mathbb{R}^{s\times s}\}$, and define the lower spectral radius (LSR) (aka joint subradius) by
$LSR(\mathcal{A}) = \lim_{n\rightarrow\infty} \inf_{A_{...
- 131
3
votes
0
answers
120
views
References on P vs NP under various axiomatic systems
I am teaching algorithms and theory of computation this semester and had the opportunity to dig a bit into the details of one way functions and the P vs NP problem.
This problem has resisted attacks ...
- 31
9
votes
0
answers
146
views
Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
0
votes
0
answers
96
views
Derivative bounds for self convolution of the spherical measure in $R^d$
While reading this article on near $L^1$ estimates for the spherical lacunary maximal function, I came across the estimate
$$ |\partial^{\gamma} (\widetilde{\sigma} \ast \sigma)(x)| \lesssim |x|^{-(1 +...
- 1
9
votes
1
answer
304
views
About the normal subgroups of Burnside groups
I was reading "On periodic groups of odd period $n\ge 1003$" of V. S. Atabekyan. He found that the Burnside group $B_n$ with $n\ge 1003$ has uncountably many normal subgroups. However, I was ...
- 91
4
votes
1
answer
227
views
Problem in Probability Theory and Functional Analysis
Let's consider the vector space V of bounded scalar functions, which includes the constant function 1. We assume that any uniform limit of a bounded monotonic sequence of functions from V also ...
0
votes
1
answer
64
views
Conditions required for the orbit of a set of positive measure to cover state space?
Suppose $(X, \mathcal{M}, \mu, T)$ is a measure-preserving dynamical system with $T$ invertible.
I am wondering what properties the dynamical system would need to have in order for the following to be ...
- 1,087
1
vote
1
answer
207
views
Is the vector bundle over a vector bundle, a vector bundle over the base scheme?
Suppose we have $E\to X$, a vector bundle over $X$ and $E'\to E$ a vector bundle over $E$. Composing the structure maps gives a smooth scheme $E'\to X$ over $X$. My question is, when is this a vector ...
- 187
1
vote
0
answers
127
views
Tangle hypothesis and ribbon category
The tangle hypothesis, when specialized to ordinary framed tangles, says that the framed tangles form the free braided category with all duals (i.e. considered as a 3-category, all the 1- and 2-...
- 1,262
8
votes
1
answer
587
views
One flip coin game
Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which.
He has \$1 with which to bet with. On ...
- 6,323
4
votes
0
answers
110
views
Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
- 531
2
votes
1
answer
107
views
Linear automorphism preserving a cone
Let $V$ be a finite-dimensional real vector space, and let $C\subset V$ be a closed convex cone, not contained in a hyperplane, and such that $C\cap(-C)=\{0\} $. Let $n$ be a nilpotent endomorphism of ...
- 38k
5
votes
0
answers
67
views
Definable pseudo-standard predicates in Internal Set Theory
Consider the usual language $\mathcal{L}=(\in, \mathrm{st})$ of Nelson's Internal Set Theory, and a unary $\mathcal{L}$-predicate $P$. For an $\mathcal{L}$-formula $\varphi$, let $\varphi^P$ denote ...
- 756
4
votes
1
answer
441
views
Large(ish) finite non-abelian subgroups of $\operatorname{GL}_n \mathbb C$ for $n>70$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SU{SU}\newcommand{\C}{\mathbb{C}}$My question is about large order finite non-abelian subgroups of $\GL_n\C$ without an ...
- 1,150
3
votes
1
answer
159
views
Sub-Gaussian concentration without the sub-Gaussian norm
A random variable $X$ is said to have sub-Gaussian tails with parameter $\sigma>0$ if
$$\Pr[|X|\geq t] \leq 2\exp(-t^2/(2\sigma^2))$$
I am interested if $X_0, X_1$ are independent, and have sub-...
- 2,183
1
vote
0
answers
72
views
Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
14
votes
2
answers
749
views
Solving the Bring quintic using the Monster?
I. Method
Hermite's method to solve the Bring quintic by functions that obey $x^8+y^8=1$ implicitly uses octahedral symmetry, while Emil Jann Fiedler's solution by the Rogers-Ramanujan continued ...
- 12.6k
9
votes
1
answer
1k
views
Is the number of varieties of groups still unknown?
A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
- 63.1k
0
votes
1
answer
46
views
Can we balance factors using the set of arithmetic sequence so as to achieve a product quality on both sides?
Stated simply the question is: given the set of an arithmetic sequence of cardinality $2N$, where $N$ is greater than or equal to $2$, is it possible to choose $N$ integers in such a way that their ...
- 1
170
votes
47
answers
34k
views
Every mathematician has only a few tricks
In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
1
vote
1
answer
102
views
Every tight $\tau$-additive finite measure is Radon
According to the 7.2.2 Theorem of the book "Measure Theory" written by V.I. Bogachev, every tight $\tau$-additive finite measure is Radon. The proof says: "The restrictions of a $\tau$-...
- 649
93
votes
9
answers
13k
views
Breakthroughs in mathematics in 2023
At the end of 2021, Johnny Cage asked about breakthroughs in 2021 in different mathematical disciplines. A similar question has been asked at the end of 2022, so it looks like Johnny Cage originated a ...
394
votes
115
answers
110k
views
Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in ...
0
votes
1
answer
158
views
Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted ...
- 23
7
votes
0
answers
264
views
$\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]$ where $W$ and $V$ are independent
Let $V$ and $W$ be independent random variables. Assume that $V$ is standard normal.
We are interested in the following limit
\begin{align}
\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]
\end{align}
...
- 671
-5
votes
0
answers
250
views
Can Cardinality Theory capture ZFC?
Cardinality Theory "CT" is a theory of sets of cardinals and links between them, only sets of cardinals can be assigned cardinalities. The links are unordered edges linking cardinals, they ...
- 11.3k
4
votes
2
answers
258
views
Waldhausen S-construction for exact categories
Let $\mathcal{C}$ be an exact category. Then, we can consider $\mathcal{C}$ as a Waldhausen category, where the cofibrations are admissible monomorphisms. By Waldhausen $S$-construction we know that $...
- 167
1
vote
1
answer
193
views
How to evaluate the following integral?
How to (analytically) calculate the following integral,
$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$
where $\langle z, \zeta \...
- 531
65
votes
5
answers
58k
views
Global character of ABC/Szpiro inequalities
In [M24] it is asserted that "considering $abc$ triples of the form $(1,p^n,1+p^n)$ for a prime number $p$ and an arbitrarily large positive integer $n$, one can verify that ABC/Szpiro ...
- 1,139
3
votes
1
answer
137
views
Normal subgroup of the geometrical fundamental group is the normal subgroup of the arithmetic fundamental group
I asked some questions on a descending lemma in Lawrence-Venkatesh 4 days ago, but it has not received any answer. I understood (2) now but I'm still confused on (1).
I want to ask a new question here....
- 131
1
vote
0
answers
136
views
Two elliptic curves with the same j-invariants
This is an interesting observation of mine when exploring moduli of elliptic curves.
Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
- 81
0
votes
1
answer
72
views
Relating the order of a polynomial to the resultant in the context of formal power series
I urgently need to understand how to begin or the complete proof of the following statement:$\DeclareMathOperator{\Res}{Res}$
While reading the paper here on page one, in the introduction, the author ...
1
vote
0
answers
75
views
Parametrized moduli spaces of semistable bundles by varying Kähler classes
Inspired by Liviu Nicolaescu's answer here, I was eager on trying to come up with examples of parameter spaces $\Lambda$, configuration spaces $\mathscr{C}$ and parametrized moduli spaces $\mathscr{M}$...
- 137
12
votes
0
answers
257
views
When do (or don't) residue fields generate the derived category of a ring?
Let $R$ be a commutative ring, and $D(R)$ its derived category of unbounded chain complexes. I'm interested in when the residue fields $k(\mathfrak{p}) = \mathrm{Frac}(R/\frak p)$ for $\mathfrak p \in ...
- 3,784
1
vote
1
answer
324
views
Want to show that this sum vanishes modulo p
Let $p\ge 5$ be a prime number, and consider the following sum:
\begin{align}
S &= \sum_{v_0 = 1}^{p - 2} \binom{p - 2}{v_0} \, \theta^{v_0 - 1}(Y) \cdot \theta^{p - 2 - v_0}(Y) \\
&+ \frac{1}{...
- 29
4
votes
1
answer
176
views
Grothendieck construction on fibred categories/stacks
This question is related to a previous question of mine, which has so far gone unanswered.
For a fixed site $\mathcal C$, the fibred categories over $\mathcal C$ form a (strict) $2$-category, see here....
- 383
3
votes
0
answers
133
views
Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139