# All Questions

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### Is there for every word a Hall set such that it factors to a few small Hall words?

Is it true that for any word $w$ there is a Hall set $H$ such that for the unique factorization of $w$ in $H$ as $w=x_1\ldots x_k$ it holds that (where the first two are just the conditions for unique ...
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### cohomological obstructions and rational points

Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions: 1) is $X(\mathbb{Q})$ an empty set ? 2) is $X(\mathbb{Q})$ a finite (non empty) set ...
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### Laplace equation between circles

I need to solve the simple Laplace equation $$\nabla^2f(r,\theta)=0$$ with boundary conditions: $$f(a,\theta)=g(\theta)$$ $$\lim_{A\rightarrow\infty}f(A,\theta)=1$$ where $a$ is a fixed real radius. ...
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### Alexeev's projective torus embeddings

I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I just started to study toric geometry. In chapter 2 in order to obtain an affine toric variety he takes ...
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### Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}$$ where ...
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### Genus of Covering Space of 3-Manifold

Let $M_g$ and $M_h$ be closed orientable 3-manifolds of genus $g$ and $h$ respectively and suppose that $M_g$ is an $n$-sheeted cover of $M_h$. Is there a formula that would allow us to compute $g$ if ...
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### A question on compact sets

Let $K\subset \mathbb{R}^N$ be a compact set. We say $K$ is "good" if the following property holds: Given a set of open neighborhoods $\{x\in U_x\subset \mathbb{R}^N\}_{x\in K}$ there exists a finite ...
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### Is an anticanonical Weil divisor in $\mathbb{Q}$-Gorestein variety Calabi-Yau?

Let $P$ be a normal, $\mathbb{Q}$-Gorestein variety with terminal singularities. Let $X \subseteq P$ be a normal, irreducible Weil divisor such that $X \sim_{\mathbb{Q}} - K_P$, that is ...
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### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime. ...
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### Geometric automorphism of free group respect to nonorientable suface

An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only ...
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### singularize the least inaccessible?

Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while ...
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### Linear intersection number and vertex covering number

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then ...
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### Algebra textbook recommandation for person already familiar with algebra [migrated]

Can you guys please recommend me a good textbook on algebra, given that I am already familiar with lots of algebra stuff and I want to revisit, and deepen my knowledge? (Little backstory: I started ...
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### smallest (sub-) sigma algebra containing a null set [on hold]

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$ What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$. I'm kind of ...
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### pushforward of universal objects along canonical morphisms of stacks

The kind of question I'm interested in has the following flavor: having two moduli stacks with one being an enhanced (i.e more data) version of the other with the natural "forgetting" map between them ...
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### An Interesting Markov Chain Mixing time question (coupling of random variables)

Consider a simple markov chain called reproduction process. Let $N>0$ be fixed. At each time $t$, there are $N$ balls, in which $x$ balls are colored white, and $N-x$ balls are colored black. We ...
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### How many points does 'the-most-point-contained-circle' contain at least?

Remark : This question has been asked previously on math.SE with receiving only a partial answer. Question : Letting $n\ge 2\in\mathbb N$, how can we find $f(n)$ such that the following two ...
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### Relations commuting with logical equivalence

A beginner's question, but still research-level (I hope): I am looking for theorems of the form, 'Relation X commutes with logical equivalence', where X is NOT just uniform substitution. Pointers to ...
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### What is the original reference for disorientations on tangle diagrams?

There are several invariants whose "natural" domain is a category of disoriented tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the ...
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### Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...
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### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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### What role does the quantum torus play in Noncommutative geometry [on hold]

Quantum torus is the associative algebra generated over a field by quasi-commuting variables $x_i$ satisfying $x_ix_j = q_{ij}x_jx_i$ for suitable nonzero scalars $q_{ij}$. What is the role played ...