All Questions
153,346
questions
5
votes
1
answer
173
views
Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
The question is simple:
For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$....
0
votes
0
answers
24
views
A question to the proof of Lemma 9 in "Multiple solutions for the Brezis-Nirenberg problem"
I'm currently puzzled by the final portion of the proof for Lemma 9 in the paper "Multiple solutions for the Brezis-Nirenberg problem"(DOI:10.57262/ade/1355867873). In particular, I'm unsure ...
- 631
2
votes
0
answers
107
views
What is the universal/fine uniformity on a topological group?
Cross posted from https://math.stackexchange.com/questions/4889335
I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\...
- 1,045
2
votes
0
answers
143
views
Hodge bundles associated to a family of complex manifolds
I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
- 21
5
votes
1
answer
116
views
Normal closure of $e_{12}$ in the congruence subgroup $\Gamma_1(p)\subset \mathrm{SL}_2(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$For an odd prime $p$, let
$$\Gamma_1(p)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in \SL_2(\mathbb{Z}):\begin{pmatrix}a & b \\ c & d\end{pmatrix}\...
- 95
7
votes
2
answers
173
views
Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?
Cover the $n$-disc with $n+1$ open sets $D^n = U_0 \cup \dotsb \cup U_n$. Suppose that $U_0 \cap \dotsb \cap U_n = \emptyset$. Suppose moreover that the cover is irredundant in the sense that no ...
- 61.5k
5
votes
0
answers
64
views
Structure of well-ordered commutative monoids
Let $(M,+)$ be a commutative monoid. Let $<$ be a well-ordering on $M$, where
$\forall a\in M,\ 0\leq a$
$\forall a,b,c\in M,\ a<b\Rightarrow a+c<b+c$
The first condition means $M$ will be ...
- 18.1k
-3
votes
0
answers
116
views
Friedrich Schur on the BCHD theorem (notes in English)
According to Sternberg in his book Lie Algebras,
The formula [the BCHD formula] is named after three mathematicians, Campbell, Baker, and
Hausdorff. But this is a misnomer. Substantially earlier than ...
- 9,931
2
votes
1
answer
132
views
Pullback morphism of a hyperplane inclusion is zero in the derived category
Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero ...
1
vote
0
answers
81
views
Injection of $G(k)/Z(k)$ into $(G/Z)(k)$
In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
- 131
0
votes
1
answer
215
views
Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 197
18
votes
0
answers
556
views
Consecutive integers of the form $2^a 3^b 5^c$
Let $\mathcal{N}$ denote the set of all products of (powers of) $2,3$ and $5$:
$$ \mathcal{N} = \{ 2^a 3^b 5^c \ : \ a,b,c \geq 0 \} \subset \mathbb{N}.$$
We use the elements of $\mathcal{N}$ to ...
- 1,762
1
vote
0
answers
68
views
Percolative process distribution not equivalent to coupon collector problem distribution
I have a process where; given a $n\times 1$ matrix initially empty, an element is inserted in it at a random position, with the possibility of repeating the insertion at a filled cell. Then, after a ...
- 61
0
votes
0
answers
57
views
VC dimension of full-dimensional closed polyhedral cone in $\mathbb R^d$
Consider a fixed set of vectors $\{x_i\}_{i\in[n]}$ in $\mathbb R^d$ and closed polyhedral cone $C = \{w \in \mathbb R^d : w^\top x_i \geq 0, \forall i \in [n]\}$ with full dimension i.e. $C$ contains ...
- 1
1
vote
0
answers
57
views
Discretization of oscillating integral
Suppose I am interested in computing
$$
I \equiv \int_0^B dx \, g(x) f(x)
$$
where $B$ is a known upper bound for the integral,
$g(x)$ is a known oscillating function and
$f(x)$ is a smooth function ...
- 33
4
votes
1
answer
109
views
Legendre's Irrationality Condition for Generalized Continued Fractions
This MathOverflow post cites that Legendre allegedly showed that given $a_{i}\in\mathbb{Z}\setminus\left\{0\right\}, b_{i}\in\mathbb{Z}$,
$$\cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}...
- 145
4
votes
1
answer
220
views
The number of irreducible characters of simple groups of Lie type
Let $S$ be a simple group of Lie type defining over $\mathbb{F}_q$ where $q$ is a power of a prime $p$ and let $\sigma$ a nontrivial field automorphism of $S$.
Set $\mathrm{C}_{S}(\sigma)$ the ...
- 551
0
votes
1
answer
104
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 341
0
votes
0
answers
32
views
Mellin transform of the volume form of a probability zonoid and its fundamental strip
Let $ L^n_+$ be the set of all $n$-dimensional nonnegative random vectors $\mathbf X = (X_1, X_2,\cdot\cdot\cdot,X_n)^⊤$ with finite and positive marginal expectations, and let $\mathbf Ψ^{(n)}$ be ...
- 171
2
votes
0
answers
94
views
Characterizing the complex structure on a non-compact Riemann surface
A consequence of Torelli's theorem is that a closed Riemann surface $X$ is determined by its period matrix. More precisely, fix $(\alpha_i)$ a basis of $H_1(X, \mathbb{Z})$ and for $J$ some complex ...
1
vote
0
answers
62
views
Congruence obstructions for three consecutive powerful numbers
Powerful number is integer $m$ such that if $p \mid m$ then $p^2 \mid m$.
Powerful numbers can be represented in the form $m=u^2 v^3$.
Erdos conjectured that three consecutive powerful numbers don't
...
- 24.2k
3
votes
2
answers
205
views
Unique "clique" of differences in $\mathbb{Z}/m\mathbb{Z}$
Are there absolute constants $0 < \epsilon < 1$ and $N \in \mathbb{N}$ such that the following holds: For every $m \in \mathbb{N}$ and every $A \subseteq \mathbb{Z}/m\mathbb{Z}$ with $\frac{\...
- 73
4
votes
1
answer
171
views
Taubes' SW$\Rightarrow$ Gr
I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?
- 853
0
votes
1
answer
75
views
Continuous modification of tangent vector fields
Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
- 341
0
votes
0
answers
106
views
Homotopy 2-Category
I am reading Riehl's notes "Higher Category from scratch". I have come across the notion of a homotopy 2-category. I have not taken any course on Homotopy Theory before but I know that the ...
- 47
1
vote
1
answer
171
views
Can there be a minimal remote cardinal?
Working in $\sf Z- Reg.$ we can have sets bigger than every well-founded set, let's label such sets as "remote". Now, suppose we'll add the axiom of existence of transitive closures, and the ...
- 9,256
4
votes
0
answers
156
views
Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,109
2
votes
0
answers
122
views
GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,681
3
votes
1
answer
210
views
Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...
- 117
0
votes
1
answer
98
views
Could I possibly exploit distinct odd primes raised to 6 to solve Exact Three Cover, when reducing it in Subset Sum?
I'm solving Exact 3 Cover, given a list with no duplicates $S$ of $3m$ whole numbers and a collection $C$ of subsets of $S$, each containing exactly three elements. The goal is to decide if there are $...
- 101
1
vote
1
answer
130
views
Are the minimal nondegenerate extensions universal?
We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...
-3
votes
0
answers
63
views
On Ideals over The Ring of Integers of a Number Field [closed]
Alrighty I really just need to run this by someone to make sure I'm not saying something crazy.
First Consider $K$ a finite extension of $\mathbb{Q}$ and $[K:\mathbb{Q}]=n$. Let us call $\mathbb{O}_K$ ...
4
votes
1
answer
217
views
Compactification of rigid-analytic varieties
Is it true that any separated quasi-compact rigid-analytic variety embeds into a proper one?
For my purpose, the base field is a $p$-adic number field.
I have seen Huber's universal compactification ...
- 91
6
votes
0
answers
97
views
Running minimum of exponential random walks
Let $\{X_i\}$ be a collection of i.i.d. Exp$(1)$ random variables. For $k \in \mathbb{Z}_{>0}$, define
$$S_k = \sum_{i=1}^k X_i$$
and note that $\mathbb{E}[S_k] = k$.
I was wondering if there is ...
- 485
-1
votes
1
answer
181
views
An equality between $\pi$ and $\Gamma$ function [closed]
Consider the following equality:
$$\sum_{n=1}^{+\infty} (-1)^{n+1} \frac{(\frac{(2n-3)!!}{(2n-2)!!})^2*\frac{\pi}{2}}{n}=\frac{\Gamma(\frac{1}{4})^2}{2\sqrt{2\pi}}-\frac{2\sqrt{2}*\pi^{\frac{3}{2}}}{\...
- 169
1
vote
0
answers
100
views
Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension
Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
- 265
0
votes
0
answers
95
views
Is there a way to find the eigenvalues of a matrix using character table?
I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...
- 1
3
votes
1
answer
86
views
Chromatic numbers realised by almost disjoint subsets of $\omega$
If $H=(V,E)$ is a hypergraph then the chromatic number $\chi(H)$ is defined to be the least cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ with $|e| \...
- 45.5k
4
votes
1
answer
269
views
Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos?
I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the ...
- 43
3
votes
1
answer
389
views
Can we have an axiom that refers to itself and the prior axioms of the theory it is an axiom of?
I know that this question is little bit imprecise, I'll try to present it in the best I can.
Can one have an axiom which is self referential with respect to itself and the theory in which it belongs?
...
- 9,256
2
votes
1
answer
149
views
Blowup formula for a morphism
Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
3
votes
0
answers
107
views
Motivic $L$-functions came from automorphic representations
Langlands in his 1978 ICM talk made a conjecture that all motivic $L$-functions should arise as automorphic $L$-functions. A part of this conjecture, namely for some Hasse-Weil $\zeta$ functions is a ...
- 131
16
votes
2
answers
623
views
Etymology of “real numbers"
I would like to know why the real numbers are called “the real numbers.” I would also like to know the meaning of “real” in the phrase “real number.”
Further questions and clarifications:
I’d like to ...
- 187
3
votes
0
answers
43
views
More than one recurrence point (Birkhoff)
Birkhoff's recurrence theorem states that for a compact metric space $X$ and a continuous function $T: X\rightarrow X$, there is a recurrence point $x\in X$; the latter means that for any ...
- 3,969
0
votes
0
answers
41
views
Is stratified Z - Infinity + there is a set as big as its powerset, consistent if NF is consistent?
The question of consistency of $\sf NF$ can be seen to be equivalent to the question of whether the theory "Stratified $\sf Z$ - Regularity - Infinity + There exists a set as big as its powerset&...
- 9,256
1
vote
0
answers
72
views
Function whose derivatives eventually vanish almost everywhere
As a takeaway of this post we have the following result.
P. Let $f:[0,1]\to\mathbb{R}$ be infinitely differentiable such that for all $x\in[0,1]$ the sequence $\{f^{(n)}(x)\}$ is eventually $0$. Then ...
0
votes
0
answers
29
views
A question about the eigenfunction method and the notion of solution - distributional solution
I have a question about how a passage was made in the calculation of passage (2.5) in the calculation below. To introduce context, the author in the paper (full work) is trying to demonstrate that ...
- 585
0
votes
0
answers
261
views
Why the sine wave of a sine wave returns a sort of cubic sine wave? [migrated]
First of all, I'm not a mathematician but a programmer interested in shaders (and thus, applied vector, trigonometric and graph maths) so bear with me.
Recently, trying to create a interpolation of a ...
- 133
1
vote
1
answer
83
views
Radial Fourier transform vs Hankel transform
I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions.
Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. ...
- 222
7
votes
1
answer
491
views
Suitable closed form for the A079501
Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position).
The sequence begins with
$$
1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, ...
- 3,452