# All Questions

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### Combinatorics of palindromic decompositions

This is sort of a companion to my question Number of trivializations of a trivial word in the free group (which in turn is motivated by my earlier question here). It turns out that that question may ...
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In a Euclidian space (Hermitian as well), say $\ell^2_n$, the following inequality holds true $$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2_n.$$ In other words, ...
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### Concise disambiguation of Voronoi boundaries

Say that $x_1,\dots,x_n$ are points in the plane, with a Voronoi diagram $V_1,\dots,V_n$. The Voronoi diagram is typically defined by $$V_i = \{x:\|x-x_i\|\leq \|x-x_j\|~\forall j\}~.$$ Is there any ...
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### A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
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### Double coset separability and the existence of vanishing sequences for surface group

Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...
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### Brocard's problem [on hold]

According to Brocard's problem $$x^{2}=n!+1 \Rightarrow (x+1)(x-1)=n!$$, where $(x+1)/2$ , $(x-1)/2$ are consecutive integers, assume one of them is z, another is (z-1). z and (z-1) have all primes ...
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### Loop defects in Walker-Wang model

My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model. Elementary excitations in the WW model can be point particles, loop defects and more ...
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### Smoothing transverse intersections

Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n$ points. Furthermore, assume that the self-intersection ...
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### Is the derived category of perfect complexes idempotent complete?

Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any ...
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### Two equivalent descriptions of a physical system yielding a non-trivial mathematical formula

First I would like to admit that this question may not be entirely appropriate for this site, but I will give it a go none the less. One often hears stories about how string dualities lead to highly ...
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### Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
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### Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...
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### Anything known about the Grundy Ordinal of Sylver's Coinage

Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia: The two players take turns naming positive integers that are not the ...
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### Pascal and Brianchon's theorems generalized for hyperbolic paraboloid

I know that giving a general version of these two theorems for quadrics can be quite tricky, but if we restrict ourselves to a verssion that holds for the hyperbolic paraboloid only it should be ...
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### One question about iteration on groups

Let $G$ be a finitely generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$ Let $\phi:H\rightarrow G$ be a ...
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### Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization. Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...
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### A property of the Frechet filter and every ultrafilter

(Joint question with Piotr Szewczak.) Definitions and notation. By filter we mean a filter on $\omega$ containing the cofinite sets at least. For a filter $\mathcal{F}$, let ...
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### A classic cardinal characteristic of the continuum in disguise?

We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer. Needed definitions may be ...
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### Are there any Algebraic Geometry Theorems that were proved using Combinatorics?

I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of Combinatorics and Algebraic Geometry, and gave some examples like the ...
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### About structure of the set of perfect matchings of $K_{n,n}$

Are there any special properties known about the set of perfect matchings of $K_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure ...
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### absolutely continuous of two probability measures

Suppose $X_t$ satisfies $$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...
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### holomorphic curves invariant by lattices

Suppose I have an entire function $f : \mathbb{C} \longrightarrow \mathbb{C}^n$ for $n \geq 1$. Let $C$ be the curve $f(\mathbb{C})$ in $\mathbb{C}^n$. Let $\Lambda$ be a lattice in $\mathbb{C}^n$ ...
### Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?
In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, ...