# Tagged Questions

This tag is used if a reference is needed in a paper or textbook on a specific result.

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### Irreducibility of family of polynomials

Consider the following family of polynomials over $\mathbb{Q}$: $$f_n = x^n - x^{n-1} - \dots - 1$$ Notice that these polynomials satisfy the recurrence $$f_{n+1} = x f_n - 1$$ I would like to ...
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### Numbers divisible only by primes of the form 4k+1

Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?
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### Higgs fields whose determinant have only simple zeros

Is the following property true for every stable holomorphic bundle of rank 2 with trivial determinant on a compact Riemann surface: The space of trace-free Higgs fields, whose determinant have only ...
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### Examples of algorithms requiring deep mathematics to prove correctness

I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course). I hope this is not too broad.
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### Regularity for a div-curl system

Let $Q = [0,1]^3$ be the unit cube in $\mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
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### Equivalence of algebraic and topological monodromy representations?

Does anyone know of a reference for the following fact? Let $M_g$ denote the moduli stack of genus g curves, let $A_g$ denote the moduli stack of abelian varieties, and let $U_g \rightarrow A_g$ ...
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### First eigenvalue for strictly convex domains

Let $M^n$ be a compact Riemannian manifold with boundary, suppose 1). $Ric(M)\ge (n-1)$ and 2). the principle curvatures of the boundary is bounded from below by $h\ge 0$. Is there any results on the ...
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### level sets portrait near a critical point

Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $O$ be an isolated critical point of $f$. I am looking at the local level sets diagram near $O$ from topological ...
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### Presheaves of Dendroidal Sets?

Are there any references available for presheaves of dendroidal sets? Seems like a natural extension of simplicial presheaves.
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### Three dimensional representations of Alternating group

The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
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### Bounds on the smallest real positive root of a polynomial

I'm trying to find upper and lower bounds of the smallest positive root of a polynomial, stated in terms of its coefficients. As I appreciate it might be a very general problem, My specific interest ...
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### distance from the mean of a normal distribution to the span of a random sample

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from ...
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### How to locate articles from Expositiones Mathematicae from 1988 and 1992

Unfortunately, my university library seems unable to find the following (I've tried the interlibrary loan tool but this particular journal is somehow outside its scope): W. C. Waterhouse, ...
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### Infinite families in stable homotopy groups

I will be very grateful for any advise or reference on the following. 1- How much is known about infinite families in ${_2\pi_*^s}$, the $2$-component of the stable homotopy ring? 2- How much is ...
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### Dye's Theorem for real von Neumann algebras

Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically ...
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### Definition of Hecke operators on orthogonal modular forms

In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5: "Describe how the correspondence in this paper behaves under the action of Hecke operators." Since ...
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### Computational complexity of computing simplicial homology

Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
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### a continuous analogue of a graph theory question

I am reading a paper and it mentions a continuous analogue of a related graph theory question that people concern. The question is that suppose $E\subset Q=[0,1]^2$ has lebesgue measure $|E|>0$, is ...
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### Higher dimensional analogs of logarithmic density

For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by \begin{align*} \underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\ \bar{d}(A)=\limsup_{N\to\...
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### Simultaneous extension of modules

Let $R$ be a commutative ring. Suppose $R$-modules $X,A,B,C$ and $Y$ are given such that the outer two rows and the outer two columns in the following diagram are exact. $\hskip1in$ Does it ...
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### Nonabelian $H^2$ and Galois descent

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety ...
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### Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$

It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise ...
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### What is known about maps between loop spaces of Spheres? - Reference request

What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...
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### Symmetries of modular categories coming from quantum groups

This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...
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### Reference Request: $M_t/M_t/1/K$ queue length distributions
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals $$4(d_1(n)-d_3(n)),$$ where the function $d_i$ counts the number ...