Trending questions
159,032 questions
3
votes
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Is there a (simple) criterion for membership to the base field of an inseparable extension?
Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$.
I'm looking for a simple way to establish if $e \in F$.
If $E/F$ ...
- 157
3
votes
0
answers
67
views
Effective action of unbounded operators on subspaces outside their domains of definition
Consider a densely defined, self-adjoint operator
$$
H: \mathcal{D} \rightarrow \mathscr{H}.
$$
Assume for simplicity that $H$ is nonnegative.
We want to effectively restrict this operator $H$ to a ...
- 133
0
votes
0
answers
49
views
The relation between Hodge bundles with metric and polarized variation of Hodge structures
Recently I've been reading Simpson's paper "constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, 1988, JAMS". On page 898 he mentioned about ...
- 11
0
votes
0
answers
42
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Ranges of operators acting on Fréchet Spaces
My question is in the spirit of Reference request: Baire's theorem for operator ranges. It mentioned that :
Finite intersections and sums of operator ranges are operator ranges.
Images and pre-...
- 101
1
vote
0
answers
41
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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
6
votes
2
answers
469
views
About Grothendieck and special cases
I was recently reminded of a quote about (not by!) Alexander Grothendieck that I had read many years ago, I think in the 1990s or 2000s.
The quote was about the way in which Grothendieck solved ...
- 887
5
votes
1
answer
323
views
Non-negative coefficients polynomials
Let $n \in \mathbb N$ and $P,Q \in \mathbb R_+[x]$.
Is it true that $(x+1)^n\neq (x-2)^2 \times P(x)+(x-4)^2 \times Q(x)$ ?
I have asked, this question here (*), two weeks ago, but no answers.
(*) ...
- 4,074
1
vote
0
answers
375
views
How to verify if sets satisfying cardinality condition exist? [migrated]
I am trying to find out if sets satisfying the following properties exist:
Call the sets $A_1, \ldots, A_{20}$ and $B_1, \ldots, B_{20}$.
For each $i \in \{1, \ldots, 20\}$, $|A_i| \in \{1,2\}, |B_i| ...
- 119
13
votes
4
answers
1k
views
Publishing corollaries of previously published results
About a year ago, I published a paper. Since then, through discussions with some colleagues, we have identified several interesting corollaries that can be derived from its results, with only minor ...
- 149
0
votes
0
answers
35
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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
2
votes
0
answers
80
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
2
votes
0
answers
112
views
Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
- 51
1
vote
0
answers
39
views
Reference request - Fourier multiplier of vector valued function
I would like to understand the concept of multiplier for vector valued functions and find appropriate references for the multiplier theorems out there.
For instance say that we would like to express $\...
- 111
20
votes
1
answer
2k
views
Can you see through a cannonball packing?
More precisely, in a regular sphere packing, either the
HCP or FCC lattice packing,
does there exist a line $L$ disjoint from every sphere,
i.e., not touching any sphere?
If so, one could "look ...
- 151k
1
vote
0
answers
58
views
Quantitative multivariate CLT from quantitative CLT of linear combinations
Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\...
- 111
3
votes
0
answers
36
views
Proving a proposition about the provability logic GL without using its completeness theorem
Let $GL$ be the provability logic containing the axioms $K := \Box (\varphi\to \psi)\to (\Box \varphi\to \Box \psi)$ and $L := \Box(\Box \varphi \to \varphi)\to \Box \varphi$, along with the ...
0
votes
1
answer
195
views
How slow can an uncomputable function from $\mathbb{N}$ to $\mathbb{N}$ grow? [closed]
I found this question here on MO: What about the fastest-growing non-computable function ?
and at first I thought I misread it. Given that all uncomputable functions seem to grow mind-bogglingly fast, ...
- 2,493
7
votes
1
answer
318
views
Example of non homogenous manifold with a finitely generated algebra of natural functions
Let $(M,g)$ be a Riemannian manifold. Let $C^{\infty}_{Nat}(M,g)$ be the $\mathbb{R}$-algebra of scalar invariants of the curvature tensor and all its higher covariant derivatives. An example of a ...
- 1,117
2
votes
1
answer
144
views
Inequality for hermitian matrices
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $\mathbf R^n, n \geq 2$ (i.e., $p_i^2=p_i=p_i^*$ and $p_1+p_2=\bf{1}$) and $S_1, S_2$ be two hermitian operators such that $S_i \...
- 73
1
vote
0
answers
71
views
Eigenfunctions of the Laplacian on $\Bbb R^d$ in Fourier space
Let $\Delta$ be the Laplacian on $\mathbb R^d$. There are no eigenfunctions of the Laplacian in $L^2(\mathbb R^d)$, but $e^{ik\cdot x}$ is an eigenfunction since
$$
\Delta e^{ik\cdot x} = -|k|^2 e^{-...
- 291
13
votes
1
answer
285
views
Finiteness of the number of Hopf subalgebras
Let $H$ be a finite-dimensional Hopf algebra over the complex field.
Question: Does $ H $ have a finite number of Hopf subalgebras?
In the case where $ H $ is semisimple, the answer is yes. According ...
0
votes
0
answers
80
views
Visual boundary vs Bowditch boundary [closed]
Is there any difference between the visual boundary of a relatively hyperbolic group and the Bowditch boundary of a relatively hyperbolic group?
Visual boundary is generally associated to a CAT(0) ...
- 1
2
votes
0
answers
84
views
How strong is separation + reflection without transitivity?
Consider a theory $T$ with a binary relation $\in$ and the following axiom schemas:
$\exists u \forall x (x \in u \leftrightarrow x \in a \land \phi)$ where $u$ is not free in $\phi$. This is the ...
- 2,213
1
vote
0
answers
169
views
Where can I find the book Haïm Brezis: Un mathématicien juif by Jacques Vauthier?
This year marks the passing of Haïm Brezis, and I would like to explore his life and work through this publication.
I tried to find it in several bookstores, in online format, but I really couldn't ...
- 509
1
vote
0
answers
53
views
Description of all biholomorphic maps from annulus [duplicate]
Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected?
In ...
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0
votes
1
answer
127
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Inequality for commuting hermitian operators
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $R^n$, $n\geq 2$ (i.e., $p^2_i=p_i=p^*_i$ and $p_1+p_2=\bf{1}$) and $S_1,S_2$ be two commuting hermitian operators on $R^n$ (i.e., ...
- 73
2
votes
1
answer
128
views
Reference request for elementary convex geometry property
I need to use the following lemma for a proof. It is an elementary result, which I am sure is well known, just I am not familiar enough with the relative literature to find a direct reference. Is ...
- 345
3
votes
1
answer
344
views
Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$
The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\...
- 48.9k
1
vote
0
answers
47
views
What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
- 168
15
votes
5
answers
3k
views
Examples of mathematical theories that are naturally written in exotic logics
Zermelo's set theory (ZF), Peano's arithmetic (PA), Tarski's theory of closed real fields (RCF), Hilbert-Tarski's geometry, etc. are all naturally expressed in first-order logic (FOL) (with a single ...
- 259
8
votes
1
answer
1k
views
Philosophy behind the Ricci flow
I don't know if my question is too simple for this forum but let me proceed.
In Ricci flow one equips a smooth manifold $M$ with a Riemannian metric $g_0$ and evolves the metric with "time": ...
- 521
0
votes
0
answers
83
views
Singular behavior of zeros of incomplete zeta function
I've been looking at the zeros of the incomplete zeta function
$\zeta_{lower}(s, z)$ recently.
$$
\zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}...
-1
votes
0
answers
42
views
Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
13
votes
2
answers
498
views
Closed form of $\frac{1}{\pi^{n}}\int_{\mathbb{R}^n}dV \prod_{i=1}^{l}\frac{1}{1+(v_{i}^{T}x)^2}$
Is it possible to find closed form of $$I=\frac{1}{\pi^{n}}\int_{\mathbb{R}^n}dV \prod_{i=1}^{l}\frac{1}{1+(v_{i}^{T}x)^2}$$
in terms of vectors $v_i$?
Where $x=(x_1,\ldots,x_{n}),\ dV=dx_1\wedge\...
- 231
4
votes
2
answers
207
views
Normalizers of the principal congruence subgroups in $\mathrm{GL}(n,\mathbf Q)$
A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence ...
- 43
4
votes
1
answer
86
views
What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
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-4
votes
0
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73
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Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?
This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
- 6,982
2
votes
0
answers
122
views
Polynomial discriminant equation
This is a fairly straightforward question, and I am hoping a definitive answer exists.
Does there exist a quadratic form $A \in \mathbb{C}[x_1, x_2, x_3, x_4]$ and a cubic form $B \in \mathbb{C}[x_1, ...
- 26.9k
1
vote
1
answer
63
views
Connection on associated bundle
Let $M$ be a compact Riemannian manifold without boundary, and consider $E$ be a vector bundle over M with metric structure on the fibers $F$. Now consider two connections $\nabla$ and $\nabla'$ on $E$...
12
votes
0
answers
243
views
+50
Is there a decidable theory of arithmetic with a non-collapsing quantifier hierarchy?
This question is very close to this old MSE question of mine, which is still unanswered.
Is there an (ideally reasonably-natural!) expansion of the structure $(\mathbb{N};+)$ in a finite language ...
- 20.5k
16
votes
5
answers
2k
views
Sets of integers with same sum and same sum of reciprocals
Let $A$, $B$ be two distinct sets of natural numbers. Is it possible to have $\ \sum_{a\in A}a\,=\,\sum_{b\in B}b\ $ and $\ \sum_{a\in A}1/a\,=\,\sum_{b\in B}1/b\ $ at the same time?
- 169
7
votes
1
answer
300
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What does Robert Stong mean when he says $H^*(MO(k))$ is a free Steenrod algebra in dimension less than $2k$?
$\newcommand{\Z}{\mathbb Z}\newcommand{\a}{\mathfrak a}\newcommand\widetildeH{\smash{\widetilde H}}$In Robert Stong's notes on Cobordism Theory, on page 95 he asserts the following:
$\widetildeH^* (...
- 391
0
votes
1
answer
82
views
Median of cardinality of set union
Let $U$ be an arbitrary finite universe (you can just think of it as $[N]=\{1,2,\ldots,N\}$), and $\mathbf{S} = (S_i)_{i \in [n]}$ ($S_i \subseteq U$) be the sets that we are drawing from. Define a ...
- 29
10
votes
1
answer
666
views
Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
- 168
6
votes
1
answer
282
views
effective descent of coherent sheaves
I am new to stacks and algebraic spaces. I have the following question:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...
- 613
2
votes
0
answers
105
views
Is this theory synonymous with ZF + Global Choice?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x \\ x \leq y \iff x < y \lor x=y \\ x \not > y \iff \...
- 11.3k
2
votes
0
answers
49
views
Reference on eigenvectors of $-\Delta $ with boundary conditions on $\Omega$
Let $\Omega\subset\mathbb R^d$ be a compact and connected subset with smooth (or piecewise smooth) boundary denoted by $\partial \Omega$. Let $\Gamma^+, \Gamma^- \subset\partial \Omega$ be such that
$$...
- 1,399
17
votes
2
answers
2k
views
Do the surreal numbers enjoy the transfer principle in ZFC?
The surreal field $\newcommand\No{№}\No$ is definable in ZFC, and it is easy to see that the surreal order is $\kappa$-saturated for every cardinal $\kappa$, precisely because we fill any specified ...
- 236k
3
votes
0
answers
157
views
Faithful representations and symmetric powers
In the question Faithful representations and tensor powers, several proofs demonstrate that for every faithful complex representation $V$ of a finite group $G$, every irreducible complex ...
0
votes
1
answer
160
views
Weak convergence of $f(x,e^{itx})$
This is the desired result (what I want to prove):
$$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$
Given that $f\in C([a,b]\times\{e^{i\...
- 231