Trending questions
159,064 questions
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
4
votes
1
answer
236
views
If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 43
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
25
votes
3
answers
3k
views
Background for the Elkies-Klagsbrun curve of rank 29
Elkies and Klagsbrun have recently announced an elliptic curve over Q of rank (at least) 29, improving on the previous record from 2006!
https://web.math.pmf.unizg.hr/~duje/tors/z1.html
It has trivial ...
- 1,139
4
votes
1
answer
438
views
Delta distribution on manifolds
Let $M$ be a manifold. There is the sheaf of distributions $\mathcal{D}'$ and the sheaf of distribution densities $\mathcal{D}(\cdot)'$ on $M$. A delta distribution density $\delta_p \in \mathcal{D}(M)...
- 341
20
votes
2
answers
3k
views
Can a 2-sphere be squashed flat?
Does there exist a function $f:\Bbb{S}^2\rightarrow\Bbb{R}^2$ which preserves the length of every rectifiable curve? That is, can a sphere be crushed flat without tears? Of course, this is a Nash-...
- 430
10
votes
1
answer
316
views
Weakly metrizable sets in normed spaces
A similar question was asked on MSE without getting an answer.
In the proof of lemma 1.2 of Asplund operators and holomorphic maps the author (my attempt to contact him failed because the only e-mail ...
- 16.5k
1
vote
0
answers
161
views
Special elliptic pencil of an Enriques surface (arguments in a proof)
I have a couple of questions about arguments in the proof of Lemma 2.6 (see absol page 199, rel p 9) from Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
The setup: Let $Y$ ...
- 5,966
1
vote
0
answers
72
views
Do almost all Gibbs' measures satisfy the weak-Poincare Inequality?
I am trying to interprete the discussion given in Section 3 of this paper,
https://core.ac.uk/download/pdf/82217936.pdf
Lets suppose we restrict to considering Gibbs's measures of the form $\sim e^{-...
- 617
4
votes
1
answer
298
views
Oscillation of monotone real-analytic function
Let $f:(a,\infty)\rightarrow \mathbb{R}$ be a real-analytic and strictly monotone function. I have been wondering how much this function can "oscillate". Namely, can we always find a ...
- 1,045
0
votes
0
answers
110
views
Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces
Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states ...
- 5,966
15
votes
1
answer
518
views
Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
- 1,336
2
votes
0
answers
79
views
determinant of cohomology of symplectic local systems
Let $X$ be an algebraic curve over $\mathbb C$ and let $E$ be a symplectic local system on $X$. For simplicity let us assume that $H^0(E)=H^2(E)=0$. Then $H^1(E)$ has a symmetric bilinear form and ...
- 7,037
3
votes
1
answer
161
views
Upper and lower bounds for a Rademacher-type expectation
Suppose that $\varepsilon_i$
are independent Rademacher random variables
(that is,
$
\mathbb{P}(\varepsilon_i=-1)
=
\mathbb{P}(\varepsilon_i=1)
=1/2
$.
Fix an $a\in\mathbb{R}^n$
and define the random ...
- 6,541
12
votes
2
answers
634
views
Can there be a proper class of Dedekind-finite cardinals?
To formalise the question: Does $\mathsf{ZF}$ prove the existence of an ordinal $\alpha$ such that, whenever $X$ is Dedekind-finite, there is an injection $X\to V_\alpha$? (Equivalently the existence ...
- 2,206
3
votes
0
answers
115
views
Extending triangulations on surfaces
Suppose that $M$ is a surface (i.e., connected topological 2-manifold, I am willing to assume compact, but possibly with boundary), $K$ a finite simplicial complex, and $f$ an embedding of $K$ into $M$...
- 3,115
6
votes
0
answers
156
views
Covering categories with posets
Let $C$ be a small (1-)category.
There is always a poset $D$ and a functor $p : D \to C$ such that:
$p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...
- 15.9k
1
vote
0
answers
69
views
Compressions and (ir)rational trace
Let $\mathcal{M}$ be a type $II_1$ factor with tracial state $\tau$ and $P$ be a projection in $\mathcal{M}$ such that $\tau(P)=1/n$ for some natural number $n.$ It is known (Ananatharaman-Popa "...
3
votes
0
answers
120
views
Signature vs commensurability
If a closed oriented $4n$-dimensional manifold $M$ has an orientation-reversing homeomorphism, then the signature $\sigma(M)$ of the intersection form vanishes. More generally, $\sigma(M) = 0$ if $M$ ...
- 41
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,974
6
votes
0
answers
554
views
What remains true after condensation?
As I slowly develop intuition for the condensed formalism, I feel that it’d help greatly if there were a principle or meta-theorem which said that certain kinds of statements that are true in ordinary ...
- 217
9
votes
3
answers
2k
views
Smallest root of a degree 3 polynomial
Is it true that the smallest root $t$ of the polynomial
$$
20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
- 171
0
votes
1
answer
86
views
Radial Lebesgue space definition
I understand that one way to define the radial Lebesgue space $L_\text{rad}^{p}(\mathbb{R}^n)$ is by the completation of the space of radial smooth function with compact support, i.e, $L_\text{rad}^p(\...
- 101
11
votes
2
answers
521
views
Knots having the same Alexander module which are not S-equivalent
As is discussed in this answer, S-equivalence class can be regarded as the Alexander module plus Blachfield pairing, an isomorphism of some duals of Alexander modules.
There are examples of knots ...
- 548
1
vote
0
answers
113
views
Analytic vector bundle from an etale local system is algebraic?
Suppose $X$ is an algebraic variety over $\mathbb C$, and $\mathbb L$ is a $\mathbb Q_p$-local system on $X_{et}$, then it corresponds to a representation $\pi_1(X_{et})\to GL_n(\mathbb L)$. Since ...
- 785
1
vote
1
answer
156
views
Determinant bundle over homogeneous varieties
I am looking for a way to compute the determinant of a homogeneous vector bundle over any homogeous variety. I am awere of how these computations work for the $A_n$ case (i.e., for flag varieties), ...
- 41
6
votes
1
answer
173
views
$\omega$-de-Bruijn sequences
Let $\omega$ denote the set of non-negative integers. For which integers $n>1$ is there a sequence $b_n: \omega\to\omega$ with the following property?
Whenever $v\in\omega^n$ there is a unique $...
- 48.9k
4
votes
1
answer
232
views
What is known about string functions for $\widehat{\mathfrak{sl}_2}$?
Let $\Lambda$ be a dominant integral weight for $\widehat{\mathfrak{sl}_2}$. The string function associated to a maximal weight $\lambda$ of $L(\Lambda)$ is the series $$ a^{\Lambda}_\lambda = \sum_{k ...
- 820
2
votes
2
answers
154
views
Closure of $C([0,1]^2)$ via weak*-topology [closed]
Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$.
The dual space of $C([0,1]^2)$, denoted by $C^*([...
- 359
2
votes
0
answers
104
views
Find an $a \times m$ submatrix of an $n \times m$ matrix with smallest rank
Given a matrix $n \times m$, I want to find the submatrices $a \times m$ by selecting $a$ columns such that their rank is minimal. Can this problem be solved efficiently?
- 1,207
3
votes
1
answer
248
views
Is the derivative of a Lipschitz function continuous a.e.?
Let $f:(a,b) \to \mathbb R$ be Lipschitz.
The derivative $f'$ exists on some set $D \subset (a,b)$ of full measure and is bounded (by Rademacher).
Is $f'$ continuous (or some representative) on the ...
1
vote
0
answers
117
views
Quotient of K3 surface: complex vs positive characteristic
Let $f: X \to X$ be a non-symplectic automorphism of finite order of complex projective K3 surface $X$. (Recall: Non-symplectic means that the induced action on $H(X,K_X)=H^0(X, \Omega_X^2)$ is not ...
- 5,966
5
votes
1
answer
268
views
What oracles make finding isomorphism (of finite structures) easy?
Below, all structures are finite, in a finite language, with underlying set an initial segment of the natural numbers. This has been edited to fix errors pointed out by Emil Jerabek in his answer ...
- 20.6k
2
votes
1
answer
192
views
Does Serre's condition $S_k$ depend only on codimension $\leq k$ points?
Recall a locally Noetherian scheme $X$ has Serre's condition $S_k$ if for every $x\in X$ we have $\mathrm{depth}(\mathcal{O}_{x,X})\geq \mathrm{min}(k,\mathrm{dim}(\mathcal{O}_{x,X}))$.
Let $X$ be a ...
- 119
1
vote
0
answers
42
views
Sub-Gaussian analysis via bounded decomposition?
Let $\psi_\alpha(x) := \exp(x^\alpha)-1$.
The Sub-Gaussian Norm $\lVert X \rVert_{\psi_2}$ of a random variable $X$ is defined as
$$
\lVert X\rVert_{\psi_2} = \inf\{c>0\mid \mathbb{E}[\varphi_2(|X|/...
- 2,183
71
votes
28
answers
8k
views
Results from abstract algebra which look wrong (but are true)
There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
1
vote
0
answers
118
views
'Uniformity' of surfaces of 3D convex solids
We try to go a little further from Which convex solids have geodesics on the surface that lie entirely in a plane?
Definitions: The surface of a finite 3D convex body may be called a convex surface. ...
- 5,979
424
votes
93
answers
149k
views
Video lectures of mathematics courses available online for free
It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
0
votes
0
answers
60
views
Tilings of $\mathbb{R}^n$ and Riemannian manifold that is uniformly locally isometric to a ball in $\mathbb{R}^n$
Suppose that we have a Riemannian manifold $(M, g)$ that is uniformly locally isometric to a ball in $\mathbb{R}^n$, that is, there exists $r > 0$ such that for every $x \in M$ ball $B(x,r)$ in $M$ ...
- 820
5
votes
1
answer
214
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
5
votes
1
answer
257
views
Does a special property hold if the Archimedean property for reals doesn't hold?
Suppose $\mathbb{R}^e=A \cup B$ in which $A \cap B=\varnothing$ and there exist real numbers $a_0$ and $b_0$ such that $a_0 \in A$ and $b_0 \in B$. My question is, can we construct $a \in A$ and $b \...
4
votes
0
answers
133
views
Ring theoretical aspects of the DAHA
The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively).
Nowdays there are many variations of the ...
- 3,318
4
votes
1
answer
120
views
Proving Equal Set Sizes in Sequential Point Selection on a Real Interval with Variable-Length Intervals
I'm here as an engineer working on a point sampling algorithm and I've noticed that when I perform the algorithm on an ordered set of points in one direction it selects the exact same number of points ...
- 151
2
votes
1
answer
208
views
Proving an exponential sum inequality for symmetric Hamming distance sequences in binary vectors
Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where ...
- 359
4
votes
0
answers
115
views
Complexity to find "short" (e.g. polynomial in diameter) decomposition of the permutation into the product of generators?
Question 1: Consider the symmetric group $S_n$ and some set of permutations $p_i$. Given permutation $g$ - what is known about the algorithmic complexity to decompose $g$ into product of $p_i$ ...
- 24.9k
2
votes
1
answer
156
views
strict inequality for Fatou's lemma
It is not the well-known form of Fatou's lemma. It is shown as below:
let $g\ge 0$ be continuous. If $X_n$ weakly converge to $X$ then
$$\lim\inf_{n\rightarrow \infty} Eg(X_n)\ge Eg(X)$$
I'd like to ...
1
vote
0
answers
50
views
Does Riemann Hilbert correspondence commute with proper smooth pushforward?
Suppose $f:X\to Y$ is a proper smooth morphism of two analytic varieties over $\mathbb C$. Let $\mathbb L$ be a local system on $X$, I want to ask do we have $R^if_*(\mathbb L)\otimes \mathcal O_Y\...
- 785
2
votes
0
answers
92
views
Lifting smooth proper varieties over finite fields to finite extensions of $W(k)[1/p]$
Let $k$ be a finite field of characteristic $p > 0$, and let $X$ be a smooth proper variety over $k$. It is generally unknown whether $X$ admits a smooth proper lifting over $W(k$, where $W(k)$ ...
- 519
1
vote
0
answers
94
views
One question about Theorem 4 in Brezis–Merle's famous paper
I'm reading Theorem 4 of the paper "Uniform Estimates and Blow-up Analysis for Solutions of $-\Delta u = V(x) e^u$ in Two Dimensions" by Brezis and Merle, and at the beginning of the proof ...
- 213
3
votes
1
answer
167
views
Finite flat maps
Let $f : A \to B$ be a finite, finitely presented, flat map of (commutative) rings. It is a known consequence of Chevalley's theorem (on constructible sets) that the induced map $Spec B \to Spec A$ is ...
- 2,040