# Questions tagged [reference-request]

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

9,746
questions

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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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18 views

### 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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8 views

### Alcove address characterization of weak order reference request

Let $\Phi$ be the root system of type $A$. Let $\mathcal{A}$ be an alcove of the corresponding affine arrangement. The address (or Shi coordinates) of $\mathcal{A}$ is a function $k:\Phi^+ \rightarrow ...

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132 views

### When is the poset of acyclic orientations of a graph a lattice?

$\def\inv{\mathrm{inv}}\def\Acyc{\mathrm{Acyc}}$Let $G$ be a graph whose vertices are numbered $\{ 1,2, \ldots, n \}$. Given an orientation $\omega$ of $G$, define the inversions of $\omega$, written $...

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38 views

### 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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59 views

### 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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91 views

### Subgroups of compact Lie groups generated by a subset of nodes of the Dynkin diagram

Where can I find a reference for the following fact, or as close as possible to it?
Let $G$ be a semisimple compact real Lie group with rank $r$, let $T$ be a maximal torus in $G$, let $\mathfrak{g}...

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58 views

### Provenance of a result on regular simplices with integer vertices

There are several MO questions related to the question of characterizing those integers $n$ for which there exists a regular $n$-simplex in $\mathbb{R}^n$ with integer vertices, e.g., coordinates of ...

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23 views

### Feasibility Criteria in Integer Linear Programming

Consider an integer linear programming problem:
For $A\in M(m, n, \mathbb{Z})$ and $b\in \mathbb{Z}^m$
find $x=(x_1,\ldots,x_n)^T\in \mathbb{Z}^n_{\geqslant0}$ such that $Ax=b$.
Sometimes one ...

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40 views

### Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...

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61 views

### Quantitive and computational improvement of the Oseledets multiplicative ergodic theorem for irrational rotation

Consider irrational rotation $T:S^1\to S^1, T(x) = x + \alpha$ where $\alpha\notin \mathbb{Q}$ (you may assume additional number theoretic properties of $\alpha$, say $\alpha = \sqrt{2}$ is already ...

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241 views

### A (surprising?) expression for $e$

I apologise if this is off topic.
Consider the quantity
$$
F(m,n,k)=\frac{(m)_k}{k!n^{k-1} }
$$
where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation
$$
\sum_{k=1}^{K} ...

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**1**answer

84 views

### Uniqueness of presentation for semi-abelian varieties

Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence
$$ 1 \to T \to G \to A \to 1$$
of algebraic groups, where $T$ is an algebraic ...

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80 views

### Is a presentation of the hyperbolic orthogonal group of rank 2 over the integers known?

The hyperbolic orthogonal group $O_{g,g}(\mathbb{Z})$ often appears in the study of high-dimensional manifolds, see e.g. work of Kreck or Galatius and Randal-Williams. Let $H$ denote the lattice $\...

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33 views

### Reference request for solving pde numerically [on hold]

What is the reference should i read to solve this pde numerical ?
$$\frac{\partial u}{\partial t} - r (\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})+\sin(y)\frac{\partial u}{\...

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23 views

### Beck-Fiala Discrepency Type Results for Arbitrary Graph Labelings

Suppose we have a graph $G$ on $n$ vertices $x_1 , \dots, x_n$ attached with weights of values from $1$ to $n$. We will write $\text{weight}(x_i)$ as simply $x_i$ and let $\text{diff}(G) = \min _{(x_i,...

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76 views

### 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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28 views

### 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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58 views

### 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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129 views

### 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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56 views

### 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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392 views

### 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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**1**answer

181 views

+100

### 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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72 views

### 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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**1**answer

152 views

### 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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141 views

### 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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82 views

### The space of $k$ differential forms as a Fréchet space

Given a smooth manifold $M$, can define define seminorms on $\Gamma(U,\bigwedge^kT^{\ast}M)$ for $U$ a coordinate open set by the following: $p^{s}_L(u = \sum_{I}u_I dx_I) = \sup_{x \in M}\max_{|I|=p, ...

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89 views

### Representation-finite quivers over dual numbers

Given a Dynkin quiver $Q$ and a field $K$.
Question 1: For which such $Q$ are there only finitely many indecomposable representations over the dual numbers $K[x]/(x^2)$?
Note that those ...

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**1**answer

461 views

### Simplicial set of permutations

Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...

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145 views

### Is there a name for a “stable” physical measure?

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support.
Definition. The ...

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155 views

### Diffeomorphism classification of Grassmannian manifolds

Is anything known about the diffeomorphism classification of Grassmannian manifolds? I know that there are some results on projective spaces (for example in Lopez de Medrano's "Involutions on ...

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**1**answer

178 views

### Tannaka duality for closed monoidal categories

I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people ...

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139 views

### Kuga-Satake in characteristic $p$ [on hold]

Have Kuga-Satake correspondences been investigated in characteristic $p$?
(I'm being intentionally vague about what this would mean.)

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163 views

### Explicit formula: explicit work with general smoothing?

The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...

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**1**answer

94 views

### Flat scalar curvature on 4 manifold

Let $(M,g)$ be a closed(oriented) Riemannian 4 manifold. It is well-known that, if $scal^g\geq0$ and not identically zero, then $M$ admits a PSC metric by conformal transformation.
Q Is $T^4$ the ...

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231 views

### Reference request for some result of de Bruijn on zeros of some holomorphic function

In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $...

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87 views

### 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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82 views

### 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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115 views

### 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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70 views

### 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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49 views

### 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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94 views

### 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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46 views

### 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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164 views

### 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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**1**answer

171 views

### Reference request: quantifier elimination test

I'm having difficulty finding this result in the standard texts.
Theorem. Let $T$ be a theory in a language $\mathcal{L}$. TFAE:
1) $T$ has quantifier elimination,
2) Whenever $M, N$ are $...

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44 views

### Posets with two partial (self-)distributive operations

Let $(X, {\sqsubset}, {\circ}, {\ast})$ be a set $X$ with a strict partial order $\sqsubset$ and two partial binary operations $\circ$ and $\ast$ such that for any $a, b, c \in X$:
$a \circ b$ and $a ...

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186 views

### 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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21 views

### 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 \...

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63 views

### Partial regularity of harmonic maps into spheres

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 ...

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### Flat base change in the complex analytic setting

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 ...