Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
9,501 questions
0
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25 views
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
votes
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? ...
1
vote
0answers
41 views
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 ...
3
votes
0answers
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 ...
3
votes
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 ...
1
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0answers
18 views
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 ...
2
votes
0answers
36 views
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, ...
2
votes
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
$$E_1=1+2V, \;\;E_n=E_{n-1}+2V\cos{[2\pi(E_1+\ldots E_{n-1})]},\;n\ge 2. \tag{...
3
votes
1answer
71 views
Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative
What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?
More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of
a function
$$...
5
votes
0answers
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) ...
4
votes
0answers
20 views
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 ...
2
votes
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.
2
votes
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 $X$, ...
1
vote
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
votes
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 ...
4
votes
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 $\...
4
votes
0answers
17 views
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 ...
6
votes
0answers
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
votes
1answer
85 views
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 (...
4
votes
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 ...
9
votes
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
votes
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
$$ \mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq ...
8
votes
2answers
467 views
Does a random sequence of vectors span a Hilbert space?
Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
5
votes
0answers
61 views
Numerics for continuity equation with Sobolev vector field
Has any work been done about numerical methods for the continuity equation
$$
\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,
$$
where $...
3
votes
0answers
40 views
Comparing different types of a PDE solutions
A few days ago I was reading the paper:
"Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system" - Feireisl, Jin, Novotny, 2012 [Arxiv].
...
4
votes
0answers
36 views
Weak estimate for difference quotient of BV function
In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
6
votes
1answer
134 views
Simplicial Objects in Additive Categories
I am looking for a reference, preferably as elementary as possible, for the following statement.
Let $X_{m,n}$ be a bi-simplicial object in an additive category $\mathcal{A}$. Then the complex $|X_{...
2
votes
0answers
175 views
Understanding the geometry of $H_{n}=\{x_i \in [-N,N]:\sum_{i=1}^n x_i = 0\}$
I am not an expert in convex geometry but if we define $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{R}$ and $S_n = \sum_{i=1}^n a_i$ I suspect that for arbitrary $N \in [1, \infty) $:
...
1
vote
0answers
51 views
Gauge structure of teleparallel gravity
I am interested in references that treat teleparallel gravity in a mathematically rigorous manner, especially in regards to it being a "gauge theory of the translation group".
The standard reference ...
2
votes
0answers
30 views
Difference quotients of solutions of ODE and PDE in Sobolev setting
In the post Difference quotient for solutions of ODE and Liouville equation, it was showed that if $\Phi$ is the solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\...
5
votes
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 ...
8
votes
0answers
94 views
+50
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 ...
2
votes
0answers
45 views
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
$$f = \int_{\mathbb R} \...
7
votes
2answers
189 views
Eulerian number identity (Reference request)
The Eulerian number $A(n,m)$ is defined as the number of permutations $\sigma \in S_n$ having precisely $m$ descents, i.e. indices $i$ such that $\sigma(i)>\sigma(i+1)$.
The wikipedia entry on ...
4
votes
1answer
80 views
Weak Lebesgue spaces and an estimate for BV functions
Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the weak $L^1$ space such that
$$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$
holds for a.e. $...
3
votes
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 ...
1
vote
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 ...
1
vote
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 ...
3
votes
0answers
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 ...
1
vote
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
votes
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 ...
5
votes
0answers
74 views
+50
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
votes
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 ...
9
votes
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 ...
11
votes
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 ...
7
votes
0answers
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 ...
0
votes
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 \...
