Newest Questions
159,054 questions
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Is this extension of n-th derivatives to ordinal-indexed derivatives trivial? [duplicate]
Let $f$ be a function defined everywhere on the real line, which is infinitely differentiable everywhere, in other words, $f$ is everywhere smooth. I define the $\omega$-th derivative, where $\omega$ ...
- 2,023
1
vote
1
answer
187
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Bound the distance between two vectors on the probability simplex
Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$
$$\sup_{x>0} \...
- 135
7
votes
2
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278
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Which pairs of conjugates of $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ generate $\operatorname{SL}(2,\mathbb{Z})$?
When do two distinct conjugates of $U := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ generate $\DeclareMathOperator\SL{SL}\SL(2,\mathbb{Z})$? The classic example is $U,L^{-1}$, where $L = \...
- 7,537
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1
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108
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Bounding $\|X_1/(X_1+X_2) - Y_1/(Y_1+Y_2)\|_p$ by the closeness of $X$ and $Y$
This question is inspired by the answer to this other question, but I have tried to make it self-contained and to zoom in on the counter-example from this answer.
Suppose $\{(X_n, Y_n)\}_{n=1}^2$ are ...
- 15
6
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1
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402
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A possible alternative model for $\infty$-groupoids
I've been studying homotopy theory on myself for quite some time now, and it is to my understanding that there's still no generally accepted definition for $\infty$-groupoids. The closest to this is ...
- 69
0
votes
1
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107
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What's the lower bound for this quantity?
Suppose $p$ is a discrete distribution with $n$ values and the random variable $x$ satisfies $\mathbb{E}_p[x] = 0$ and $|x| < \infty$. Given $\alpha \in (0,1)$, does there exist a lower bound for ...
- 49
2
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1
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259
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Explicit solution to linear SDE with correlated Brownian motions
Let $W$ and $B$ be correlated one dimensional Brownian motions with constant correlation coefficient $r \in (-1, 1)$, that is, we have $d\langle W, B \rangle_t = r \, dt.$ We assume we have $B_0 = v$ ...
- 6,313
5
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1
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615
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Is every character of the algebra of continuous functions on a locally compact space some evaluation?
Given any locally compact Hausdorff space $X$, let $C(X)$ denote the complex algebra of all complex-valued continuous functions on $X$.
Question. Given an arbitrary character (i.e. a non-zero ...
- 960
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134
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Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1
Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
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1
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95
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Detecting points inside the convex hull with inner products
Given a finite set $P$ of $n\gt d+1$ points in $d$-dimensional euclidean space.
Under the assumption that the points of $P$ are in general position in the sense that $\lbrace p_{i_1},\dots,p_{i_d}\...
- 13.2k
4
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456
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An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?
Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...
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9
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1
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371
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For which subgroups the transfer map kills a given element of a group?
$\newcommand{\ab}{{\rm ab}}
\newcommand{\ord}{{\rm ord}}
$Let $G$ be a finite or profinite group. Consider the abelianized group
$$G^\ab=G/G'$$
where $G'$ is the commutator subgroup of $G$.
Let $H\...
- 14.2k
4
votes
1
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516
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Weak convergence (probability theory) and weak* convergence (functional analysis)
Let $(\mu_n)_{n\in\mathbb N}$ be a sequence of probability measures on $\mathbb R$ with the usual Borel $\sigma$-algebra $\mathcal B(\mathbb R^p)$. That is, $(\mu_n)_{n\in\mathbb N}$ can be considered ...
- 103
3
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1
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264
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(non)reduced stabilizer scheme
A well known open question is whether the scheme of commuting pairs in a complex reductive group $G$, for example in $G=GL(n)$, is reduced. The variety of commuting pairs is a special case of a more ...
- 1,526
2
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1
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If $F$ is a prosoluble subgroup of a free profinite product $\amalg G_i$ and $F \cap G_i^g$ is pro-$p$, is also $F$ pro-$p$?
There is a 1995 paper (Manusc. Math., DOI link) of Florian Pop where he proves the following:
Theorem. Let $G$ be the free product of profinite groups $G_i$ and $F$ a closed prosoluble subgroup of $G$...
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4
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1
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291
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Structure of all Wightman QFTs
I have two related questions related to constructive/axiomatic QFT.
Is there a structure on the collection of all QFTs, as defined by the Wightman axioms? Do they form some type of category?
...
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9
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147
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degree 1 maps for bordism homology
Let $f\colon X \to Y$ be a degree 1 map between closed oriented manifolds. Then the induced homomorphism between the homology groups is surjective up to torsion.
Can one say something similar about (...
1
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1
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146
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Upper bound on difference of correlated ratios
Suppose $(x_n, y_n)$ are i.i.d. samples (that is, $x_n$ and $y_n$ are not independent, but $(x_n, y_n)$ is i.i.d. with regards to $(x_m, y_m)$ if $n\ne m$) from a joint distribution, with $0 < x_n, ...
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1
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82
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How to calculate the Integral with confluent hypergeometric function
How to prove this.Thank you in advance
Let $\delta,\beta>0$ How to prove this
\begin{align}
& \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...
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1
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249
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Structural description of Bohr sets in $\mathbb{Z}_N$
Definition 1. Let $\Gamma\subset \widehat{G}$ and $\delta\in [0,2]$. The Bohr set with frequency set $\Gamma$ and width $\delta$ is the set $\text{Bohr}(\Gamma; \delta)= \big\{x\in G: |\chi(x)-1|\leq \...
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284
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Is there an $X$ and a non-surjective continuous function $f:X\to X$ such that $f\simeq {\rm id}_X$ but $f(X)$ and $X$ are not homotopy equivalent?
Let $X\subset \mathbb{Z}^n$ be a finite set (vertex set) and $1\leq u\leq n$. For $x=(x_1 ,\ldots ,x_n)\neq y=(y_1 ,\ldots ,y_n) \in X$, we define the adjacency $\kappa_u$ on $X$ as follows: $x\sim_{\...
- 1,182
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419
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How to select suitable journals for submission? [closed]
I think this post should be made community wiki, as there may not be some definitive answer, and it's more like a discussion.
My question is actually more subtle than the title of the post. What I ...
- 1,173
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131
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Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
- 64k
2
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143
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Hypercomplex structures and tangent space decompositions
For any almost complex manifold we have a decomposition of its tangent space into two subspaces $T = T^{(1,0)} \oplus T^{(0,1)}$. For an almost hypercomplex manifold we have three almost-complex ...
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134
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Isom-functor for generalized elliptic curves is representable
I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,
(page DeRa-61) (*) For $C_i$, ...
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3
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124
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$\mathbf{E}_n$-algebras in nerves of 2-categories
In Example 5.1.2.4 of Higher Algebra, Lurie explains how there is a bijection between equivalence classes braided monoidal structures on one-category $\mathcal{C}$ and $\mathbf{E}_2$-algebra ...
- 5,711
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91
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Positive semidefinite maximum principle
Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. ...
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370
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What is the nerve of this category?
If $\mathcal{C}$ is a thin category, we call $U \subseteq \mathrm{Ob}(\mathcal{C})$ open if for every object $X \in U$ and any morphism $X \to Y$, we also have $Y \in U$. This declares an Alexandrov ...
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552
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Largest group table with all real irrep dimensions different
Take for example the two groups $T$ and $I$. (See character tables - unfortunately chemists -like me- and mathematicians use different notation.) As you see, $T$ has three real irreps, and their ...
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386
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For which set $A$, Alice has a winning strategy?
Cross-posted from MSE: https://math.stackexchange.com/questions/4775193/for-which-set-a-alice-has-a-winning-strategy
Alice and Bob are playing a game. They take an integer $n>1$, and partition the ...
- 1,462
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1
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773
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How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?
We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e.,
$$
p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d,
$$
and define the operator $P_t$ by
$$
...
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3
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393
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Why is the "wrong" definition of intersection of varieties the "right" one for generalized Bézout?
For ease of notation, define the degree of a variety to be the sum of the degrees of its irreducible components. The generalized Bézout theorem (due to Fulton and Macpherson) states that, for $V_1$, $...
- 20.2k
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407
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Can Set Theory be turned into Infinite Arithmetic?
The following system I'd label as "Infinite Arithmetic" is simply an endeavor to extend second order arithmetic to the infinite ordinal world, and extending with it the representation of ...
- 11.3k
0
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1
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128
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Rational approximation for continuous function on curve $\Gamma$
Let $\Gamma \in C^{1,\lambda}$ be an oriented Jordan curve in complex plane $\mathbb{C} $, $\mathrm{R}(\Gamma)$ the set of all rational functions without poles on $\Gamma $. "$\mathrm{R}(\Gamma)$...
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240
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The most powerful inner model and a $\Delta^2_1$ well-ordering of the reals
With the current research, it seems that we are in a position to get extremely powerful absoluteness theorems (like $\Sigma^2_0$-absoluteness, $\Sigma^2_1$-absoluteness, $\Sigma^2_2$, $\diamondsuit_G$,...
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150
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Analytical Dold-Thom
Let's $X$ be a projective smooth variety over a field that has an embedding into $\mathbb{C}$. Let's denote the infinite symmetric power of $X$ by $\text{Sym}^{\infty}(X)$. Denote the algebraic ...
- 5,901
3
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311
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Heat kernel of left-invariant metric on 3-sphere
This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
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43
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Multiplicity one for characters in left regular representation of an infinite discrete group
I know not much is known about the left-regular representation of non-compact groups, but I was wondering the following: if we have a subquotient
$$R\hookleftarrow M\twoheadrightarrow \mathbb{C}(\chi)$...
- 2,054
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155
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Reducing subspaces of unitary operators
Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
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2
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730
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Concrete representation of coend in linear algebra
$\require{AMScd}$Teaching coend calculus to a PhD student led me to this "elementary" computation that I would like to perform explicitly.
Consider the functor $F : (\mathbb N,\le)^\text{op}\...
- 13.6k
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120
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Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
- 227
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116
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Sudden drop in complexity class due to the more general correlations
Recently I was asking about the impact of the groundbreaking result MIP*=RE on logic and proof theory (see this discussion). Surprising as it is I got confused with the following: MIP* is a ,,quantum''...
- 9,340
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119
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Continuity of a minimizing measure w.r.t a parameter
Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$.
My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
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2
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497
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Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?
For $U$ a unitary $N \times N$ matrix, randomly distributed according to Haar measure, we have the complex-valued random variable $\log (\det (1-U))$. The real part and imaginary parts of $\log (\det (...
- 149k
2
votes
1
answer
269
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Inner model for KP and a Well-Ordering of the Reals
It is well known that Gödel proved the following theorem:
$\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
So:
Is there an inner model for KP/Z/....
- 695
3
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1
answer
213
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Reference Request for "Finite Semilattice with Top and Bottom is a Lattice"
Let $\mathcal{O}(P)$ be a finite, completely distributive lattice of all lower sets ordered by set inclusion.
Moreover, let $K =\; \mathrel{\{} h(x) \mathrel{|} x \in \mathcal{O}(P) \mathrel{\}}$ be ...
- 33
9
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2
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599
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Is a local system on a surface determined by simple closed loops?
Let $\Sigma$ be a closed oriented topological surface of genus $g\geq 2$, and let $\mathfrak{X}_n$ denote the $\mathrm{SL}_n$-character variety of $\pi_1(\Sigma)$, i.e.
$$
\mathfrak{X}_n= \mathrm{Hom}(...
- 254
1
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0
answers
116
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Help to understand the geodesics in $BS(1, 2)$
I would like to understand the sets of geodesics in $BS(1, 2)$, which is described in https://arxiv.org/pdf/1908.05321.pdf, Proposition 3 (page 3).
Write $$ G=B S(1, 2)=\left\langle a, t \mid t a
t^{...
- 823
1
vote
1
answer
344
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Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
- 612
8
votes
2
answers
1k
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Natural density of the set of simple numbers
Let us call $n>1$ simple if every prime power $q$ with $q-1 \mid n-1$ is a prime number. (Please let me know if there is already an established name for these numbers.) The simple numbers $\leq 100$...
- 63.1k