# Questions tagged [reference-request]

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

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### Existence of algebraic integers with certain properties

Is the following statement true? ($\star$) Given integers $n > k > 0$, there exists a monic polynomial of degree $n$ with integer coefficients and constant term $\pm 1$ whose roots are all ...
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### Distributional continuity of information theorems

The minimum entropy-rate $R$ for encoding a source $X$ to within mean squared error $\sigma^2$ given access to side information $Y$ is given by the Wyner-Ziv theorem: \begin{equation} R=\min_{U:E[(X-...
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### Reformulation as optimization on probability distributions

This is a "soft" question, in the sense that I'm looking for historical remarks and general commentary rather than a definite answer. For compact $X \in R^n$ and $f : R^n \to R$ consider the problem ...
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### Irreducible skew polynomials over an algebraically closed field

Let $\mathbb{F}$ be a field, and denote with $\mathbb{F}[t,\sigma]$ the skew-polynomial ring, where $\sigma$ is an automorphim of $\mathbb{F}$. Recall that the multiplication of this ring is defined ...
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### Independent conditions imposed by a collection of double points

Let's consider the following statement : There exist a collection of $d$ points $\gamma \subset \mathbb{P}^{n}$, so that $h_{\Bbb P^n}(\gamma^{2} ,m) = \min\{(n+1)d, \binom {n+m}{n}\}$ implies for any ...
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### Neumann equation on manifold with edge or corner

Let $(M,g)$ be a compact Riemannian manifold with boudnary and corner, i.e. locally mdoelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$. ...
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### Defining pull-back of Chow groups under a morphism of special type

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. Let $\pi : Y\rightarrow X$...
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### Mapping Problems to Boolean Formulas for SAT Solvers

I came across Marijn Heule and Oliver Kullmann's paper on recent techniques in highly efficient SAT solvers. In particular they describe the Pythagorian Triple Problem, which they solved using that ...
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### Spectra of one dimensional Schrodinger operators [on hold]

I am trying to understand how to compute the spectra of one-dimensional Schrödinger operators $$\mathcal{L}:=-\partial_x^2+V,$$ where $V$ is a bounded function in the whole line. I am particularly ...
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### Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie

I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of ...
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### Symmetric monoidal category with trivial switch morphisms

Is there a specific terminology for a symmetric monoidal category in which for any object $x$ the switch map $x\otimes x\to x\otimes x$ is the identity ? (Or alternatively the action of the symmetric ...
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### On semicontinuity of Hilbert function for a zero dimensional scheme

Let, $X$ be a zero dimensional subscheme of $\Bbb P^n$ and let us define a function as follows: $h_{\Bbb P^n}(X ,d) = \binom {n+d}{n}$ - dim $I_{X}(d)$ , where dim $I_{X}(d)$ = the dimension of ...
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### For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?

Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups? I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
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### Reference request: A knot is tame if and only if it has a tubular neighbourhood

Definitions: A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal). Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...
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### Hermitian sublattices of a given type

Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-...
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### On earlier references for $P=BPP$ and Kolmogorov's possible view on modern breakthroughs involving randomness?

Kolmogorov and Uspenskii in this paper 'http://epubs.siam.org/doi/pdf/10.1137/1132060' speculate $P=BPP$ in $1986$. They do this without getting into circuit lower bounds and from a different view ...
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### Proof of Isoperimetric Inequality using Curve Shortening Flow

I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was ...
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### Literature about solitons and Hirota derivatives

This summer I'm going to learn a mini-course about soliton theory ("Soliton equations and symmetric functions" in LHSM (Russian summer school in mathematics). The web-page of this course is https://...
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### Random graphs - multiple giant components

For a given random graph, a connected component that contains a finite fraction of the entire graph’s vertices is called giant. A well known result in random graphs is the existence and uniqueness of ...
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### Computing Chow group of a variety which is almost a blow-up of another variety

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. I have a morphism which is ...
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### Reference request ( Conductor of Galois representation associated to Dirichlet character)

(Sorry for my poor english...) Let $\chi$ be a Dirichlet character modulo $N$ and $\Psi_{\chi}$ be an one dimensional Galois representation such that \begin{equation} \Psi_{\chi}: \text{Gal}(\...
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### Complex factorization of the angular part of the Laplacian

Some time ago some research led me to the following equality: \begin{equation} \frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...
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### Adequate equivalence relations and algebraic $K$-theory

I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...
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### Convergence rate of cardinal series (Whittaker-Shannon interpolant)

Given $f \in C^{k}_{0}[a, b]\cap L^{2}(\mathbb{R})$, what can we say about the convergence rate of the cardinal series  s(t) = \sum_{j=0}^{n-1} f(a+jh) \mathrm{sinc}\left(\pi\left(\frac{t-a}{h} -j \...
Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension": Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of ...