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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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Connected transitive group action and wreath product

It is well known and not hard to prove that the wreath product of two finite transitive group actions is again transitive. Apparently a stronger statement is true: suppose that $G$ and $H$ act ...
5
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0answers
24 views

Tartar wave cone — oscillations and ellipticity

In a work of De Philippis and Rindler, I found the following passage. Questions. How did they show that $\Lambda_{\mathscr A}$ contains all the amplitudes along which the system is not elliptic? ...
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Reference request: Representing posets by integer divisibility

Does anyone know of an early published reference for the (very easy) fact that all finite posets can be represented as the poset of divisibility of a finite set of integers? Page 1 of Birkhoff's ...
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61 views

Lusztig's completion for universal enveloping algebra

In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
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0answers
86 views

Global to local principle for f.g. $\mathbb{Z}[x]$ modules

In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...
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Comparing solutions of the PDE problems - one problem originates from the other

First, let's introduce the main problem. Here this will be the Cauchy problem given in the conservation form: $$ (1) \hspace{0.5cm} u_t (x,t)+ div f(u(x,t))=0, $$ where the initial condition is ...
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Factorization of characters

Linked to the end of this question here and because the subject involves many deformations of shuffle, I came to the following Let $k$ be a $\mathbb{Q}$-algebra and $\mathfrak{g}$ a $k$-Lie algebra, ...
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1answer
63 views

Cyclotron sequences

If one considers how the particle's energy grows in the (idealized) cyclotron, one gets the following sequence of numbers $$E_1=1+2V, \;\;E_n=E_{n-1}+2V\cos{[2\pi(E_1+\ldots E_{n-1})]},\;n\ge 2. \tag{...
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1answer
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Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative? More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of a function $$...
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35 views

Entropy solution for linear transport equation

Consider the transport equations $$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$ and $$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$ Can we define a notion of entropy solutions for (1) ...
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Existence of multiple entropy solutions

Consider the conservation law $$(\ast) \begin{cases} \partial_t u + \partial_x f(u) = 0, & (t,x) \in (0,T)\times \mathbb R \\ u(0,x) = u_0(x), & x \in \mathbb R \end{cases}$$ Under what ...
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1answer
43 views

Confluent partial orders

Let $(P, \le)$ be a poset such that $$ \forall a, b, c \in P: b \ge a \le c \implies \exists d \in P: b \le d \ge c. $$ I am looking for literature where such confluent partial orders are studied.
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0answers
50 views

Two definitions of horofunction for Gromov hyperbolic spaces

Let $X$ be a proper, geodesic, $\delta$-hyperbolic metric space (e.g. a hyperbolic group), and let $x_0$ be a basepoint for $X$. There seem to be two different definitions of "horofunction" for $X$, ...
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1answer
32 views

Probability that maximal elements has the same position in samples from correlated random variables

Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x_1, x_2, ..., x_L\}$ and $Y= \{y_1, y_2, .. y_L\}$ be samples of size $L$ ...
3
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1answer
56 views

Kolmogoroff condition for truncated random variables

Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...
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0answers
21 views

Failure of entropy condition for a singular limit of higher order regularization for a conservation law

Consider a regularization of the conservation law $$\partial_t u + \partial_x f(u) + \epsilon \partial_{x}^3 u = 0$$ How does one prove that the limit function $u$ of $\{u_\epsilon\}_\epsilon$ as $\...
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Existence and uniqueness of entropy solutions for a scalar conservation law

Consider the conservation law $$(\ast) \qquad u_t + \partial_x(u^\alpha) = 0$$ where $\alpha > 0$. For what values of $\alpha$ is it known that there exists a (unique) entropy solution for the ...
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54 views

Wave equation with porous medium term

The classical porous media equation is $$u_t + \Delta(u^m) = 0 \quad m>1.$$ Has the (degenerate) wave equation $$u_{tt} + \Delta(u^m) = 0$$ been subject of studies? What would the physical ...
8
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1answer
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BV functions and wave equation

What is the role played by BV functions in the study of (possibly nonlinear) wave equations? I think that one would need assume small initial data in $L^1$ or $H^1$ to get a well-posedness result (...
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0answers
72 views

moduli space of toric structures on a fixed toric variety (reference?)

I'm looking for a reference on the following question: Given a fixed toric variety $V/k$, how to describe the moduli space of all toric structures on $V$? In addition to the general question, I ...
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1answer
152 views

Reference requests: Integral cohomology of $G_2$-homogeneous spaces

Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed? Great computational efforts using representation theory in order to determine the ...
8
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1answer
192 views

An isoperimetric-type inequality inside a cube

I am looking for a reference for the following inequality: if $\Omega \subset [0,1]^d$ satisfies $\mbox{vol}(\Omega) \leq 1/2$, then $$ \mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq ...
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2answers
467 views

Does a random sequence of vectors span a Hilbert space?

Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
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0answers
61 views

Numerics for continuity equation with Sobolev vector field

Has any work been done about numerical methods for the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ where $...
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0answers
40 views

Comparing different types of a PDE solutions

A few days ago I was reading the paper: "Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system" - Feireisl, Jin, Novotny, 2012 [Arxiv]. ...
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0answers
36 views

Weak estimate for difference quotient of BV function

In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
6
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1answer
134 views

Simplicial Objects in Additive Categories

I am looking for a reference, preferably as elementary as possible, for the following statement. Let $X_{m,n}$ be a bi-simplicial object in an additive category $\mathcal{A}$. Then the complex $|X_{...
2
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0answers
175 views

Understanding the geometry of $H_{n}=\{x_i \in [-N,N]:\sum_{i=1}^n x_i = 0\}$

I am not an expert in convex geometry but if we define $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{R}$ and $S_n = \sum_{i=1}^n a_i$ I suspect that for arbitrary $N \in [1, \infty) $: ...
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51 views

Gauge structure of teleparallel gravity

I am interested in references that treat teleparallel gravity in a mathematically rigorous manner, especially in regards to it being a "gauge theory of the translation group". The standard reference ...
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30 views

Difference quotients of solutions of ODE and PDE in Sobolev setting

In the post Difference quotient for solutions of ODE and Liouville equation, it was showed that if $\Phi$ is the solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \...
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1answer
75 views

Comparison of Rademacher processes

Suppose that $T$ is a bounded set in $\mathbb{R}^n$ and $f,g$ are two nonnegative functions such that $0\leq f(x)\leq g(x)$ for all $x\geq 0$. Let $\epsilon_1,\epsilon_2,\dots,$ be a Rademacher ...
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Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
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Coarea-like formula for BV functions (not their derivative)

Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Unfortunately, the formula $$f = \int_{\mathbb R} \...
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2answers
189 views

Eulerian number identity (Reference request)

The Eulerian number $A(n,m)$ is defined as the number of permutations $\sigma \in S_n$ having precisely $m$ descents, i.e. indices $i$ such that $\sigma(i)>\sigma(i+1)$. The wikipedia entry on ...
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1answer
80 views

Weak Lebesgue spaces and an estimate for BV functions

Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the weak $L^1$ space such that $$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$ holds for a.e. $...
3
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1answer
243 views

Preparation for GIT (Geometric Invariant Theory)

I am trying to read Mumford's Geometric Invariant Theory, however, I find my knowledge in algebraic geometry is inadequate. My knowledge is at the level of Hartshorne's Algebraic Geometry. Mumford ...
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1answer
69 views

Koszul -regular sequences (reference request)

Let $R$ be a ring and let $f:P\rightarrow P'$ be a surjective morphism of smooth $R$-algebras. Let $J$ be the kernel of this map. If $R$ is Noetherian, one can show that $J$ is locally generated by a ...
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1answer
41 views

Determinants in Jordan algebras of Euclidean type

As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type. Each (some?) of such algebras admits a cone of positive definite ...
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94 views

Liftings and splittings (reference request)

I'm writing a paper and, at a certain point, I need the following, rather elementary Lemma. Assume that we have a commutative diagram of short exact sequences of groups of the form Then the ...
7
votes
1answer
159 views

Classification of 2-types — crossed modules vs. Postnikov data?

A few questions about the equivalence between 2-types and crossed modules. For simplicity, assume everything is connected. What is the precise statement? Is there an equivalence of categories (or at ...
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0answers
87 views

Mathematical background required to learn about sheaves

Due to my interest in type theory (and higher type theory), I have found that learning about sheaves might be useful (for, e.g., sheaf models of type theories). There is Kashiwara and Schapira's ...
3
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1answer
93 views

A reference for a (folklore?) characterization of K-analytic spaces

I am writing a paper on K-analytic spaces and need the following known characterization. Theorem. For a regular topological space $X$ the following conditions are equivalent: (1) $X$ is a ...
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Partially BV vector fields and renormalization

Why does the approach used to prove Theorem 4.1 in the paper by Le Bris and Lions on Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications not work ...
6
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1answer
121 views

Non-zero winding number on a space curve implies a linked curve in the zero set?

The following statement has been largely improved from my original post thanks to discussions with @DmitriPanov ad well as the comment from @Wojowu. Let $f \colon \mathbb{S}^3 \to \mathbb{R}^2$ be ...
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0answers
110 views

Where is it shown that homotopy sheaves form a higher stack?

Many references on infinity categories etc. advertise that one application is that it's an appropriate setting to glue (the appropriate replacement for) derived categories of sheaves. What's the ...
1
vote
1answer
285 views

Variation on Sylow Theorems [closed]

Does a finite group on $2^t$ elements (with $t$ a positive integer) necessarily have a subgroup of index two? It seems close to the Sylow Theorems but not quite. Maybe there is a simple ...
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2answers
1k views

Where are Serre’s lectures at Collège de France to be found?

Having run into several references, at various places and occasions, to "Serre’s Course at Collège de France, 19xy-19xy+1" for various values of xy, I would genuinely like to know where these lectures ...
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126 views

Making the precaliber number bigger than all Knaster numbers

Write $\mathfrak m_k$ for the Martin's axiom number for $k$-Knaster, i.e., for the smallest size of a family of dense subsets of some $k$-Knaster poset for which there is no generic filter. (A poset ...
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0answers
34 views

On divisibility conditions implying local coprimality conditions

This question is inspired by Bernardo Recaman's question Strings of consecutive integers divisible by 1, 2, 3, ..., N on intervals of $n$ integers being divisible by the integers $1$ through $n$. The ...
4
votes
1answer
103 views
+50

Growth assumption and example of finite (arbitrarily small) time blow up for ODE

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \...