# 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

**0**

votes

**0**answers

47 views

### Relationship between the vortex filament equation and the transport equation

Let us consider the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$.
How is the Cauchy problem for the ...

**0**

votes

**1**answer

66 views

### Relationship between the vortex filament equation and the cubic Schrödinger equation

How is the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
related to the cubic Schrödinger equation?
Note 1. ...

**1**

vote

**0**answers

40 views

### Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...

**8**

votes

**2**answers

254 views

### Hopf algebra kernels vs. algebra kernels

Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field. Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the ...

**2**

votes

**0**answers

49 views

### Criteria for a limit to be a proper function

This question is obviously broad; turning this broadness into something sharp is part of the problem.
Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...

**3**

votes

**1**answer

49 views

### Reference Request: $L^p(x)$/(Musielak–Orlicz space) analogue of classical $L^p$ result

Fix a non-empty open domain $\Omega\subseteq \mathbb{R}^d$ with compact closure, and a finite Borel measure $\mu$ on its closure $\overline{\Omega}$.
In Halmos' book it is shown that:
Classical ...

**2**

votes

**1**answer

95 views

### Surveys/monographs on the vortex filament equation

Where can I find surveys on the mathematical aspects of the vortex filament equation?
In particular, I'm interested in the following topics:
physical motivation;
notion of solutions and ...

**4**

votes

**1**answer

81 views

### Volumes of double cosets $KtK$

Let $G$ be the group of rational points of a connected reductive group $\mathbb G$ defined over a non-archimedean local field $F$. For simplicity sake, I assume that $\mathbb G$ is split over $F$. Let ...

**1**

vote

**2**answers

87 views

### Reference request: lower sets of a preorder form a lattice

Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...

**2**

votes

**3**answers

154 views

### Every linear topological space embeds into the Tychonoff product of linear metric spaces

I need a reference to the following (known?)
Fact. Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of ...

**2**

votes

**0**answers

79 views

### Open sets on a Stone space

If $B$ is a Boolean algebra (possibly assumed complete), is there a standard name for the Heyting algebra (or frame) $L := \Omega(S(B))$ of open sets on the Stone space $S(B)$ of $B$, — or for the ...

**3**

votes

**0**answers

86 views

### Box counting dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it?
The analogous question for the ...

**4**

votes

**5**answers

209 views

### Peculiarities in low dimensions or low order or etc

I have been pondering about certain conjectures and theorems viewed as either low vs high dimensions, or smaller vs larger primes, or anything of the sort "low vs high order". Let me mention a couple ...

**6**

votes

**0**answers

136 views

### What is the name for this type of families?

Is there a common name for a family $\mathscr{F}$ which satisfies the following condition?
For any infinite $X\subseteq\mathscr{F}$ there exists a finite $A\subseteq X$ such that $\bigcap A$ is ...

**1**

vote

**2**answers

98 views

### How to use probability to find a matching in a family of graphs?

In a conference, I heard that we can use some probabilistic methods to find a matching in some kind of graphs. I would like to see some examples of such technics. Can someone provide some references ...

**5**

votes

**0**answers

80 views

### Generalisation of the Witt–Berman induction theorem

$\DeclareMathOperator{\Aut}{Aut}\DeclareMathOperator{\Ind}{Ind}$I believe I can prove the following induction theorem (modulo carefully checking a few details), and I would like to know whether this ...

**1**

vote

**1**answer

99 views

### Harmonic functions vanishing on the boundary and distance function asymptotics

Let $\Omega \subset \mathbb R^N$ be a $C^2$ domain. Let $u$ be a function such that $u \in W^{2,2}(\Omega)$ and $u = \Delta u = 0$ on $\partial \Omega$. Is it true that $$ c \le \frac{u}{[\mathrm{dist}...

**2**

votes

**0**answers

87 views

### What are all pairs $(R,M)$ of a ring $R$ and a two-sided $R$-module $M$ such that all endomorphisms of $M$ are scalar multiples of $\text{id}_M$?

I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ ...

**5**

votes

**3**answers

384 views

### Existence of a weight of a representation in the fundamental Weyl chamber

Let $\mathfrak g$ be a complex simple Lie algebra.
Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system.
Pick a partial order on $\mathfrak h$, ...

**2**

votes

**0**answers

91 views

### A new topology on the dual of a locally convex space?

Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...

**3**

votes

**0**answers

77 views

### Reference for “holomorphic contact geometry”

Just like holomorphic symplectic geometry is a complexification of real symplectic geometry, I am wondering is there any good survey paper or book talking about holomorphic version of real contact ...

**7**

votes

**0**answers

206 views

### Regularity result for the boundary value problem for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$.
Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$
Consider the following boundary value problem for the heat equation:
...

**21**

votes

**7**answers

2k views

### Good introductory book to type theory?

I don't know anything about type theory and I would like to learn it.
I'm interested to know how we can found mathematics on it.
So, I would be interested by any text about type theory whose angle ...

**1**

vote

**1**answer

152 views

### Reference for “topological affine spaces”

I am wondering if there is a topological version of affine spaces as a topological space along with a free transitive (continuous) action of a topological vector space on it?
Here is a notion so-...

**0**

votes

**0**answers

75 views

### Topological entropy of logistic map $f(x) = \mu x (1-x)$, $f:[0,1] \to [0,1]$ for $\mu \in (1,3)$

As stated in the question, I want to find the topological entropy of the logistic map on the interval $[0,1]$ for a "nice" range of the parameter $\mu$, namely $\mu \in (1,3)$. I think the fact that $...

**9**

votes

**2**answers

458 views

### Involutions in $\mathbb{F}_p[[x]]$

Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$?
Here involution in $A[[x]]$ means $f\in ...

**1**

vote

**0**answers

307 views

### Reference for boundedness and constructibility of $Ri^! \mathcal{E}$

Here David Hansen says
By hard results of Deligne, Grothendieck, and Gabber, $Ri^! \mathcal{E}$ is still bounded and constructible...
What are these results of Deligne, Grothendieck and Gabber? ...

**4**

votes

**1**answer

58 views

### Convex Hull of Outer Products of (Normalised) Nonnegative Vectors

If I define $\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \| x \|_2 \leqslant 1 \}$, then (assuming I recall correctly) it is known that the convex hull of $\mathcal{A}$ is given by
\begin{align}
\...

**1**

vote

**0**answers

34 views

### Reference for tensor multiplication and derivatives from a computational / concrete standpoint

I am looking for a reference for some fairly elementary definitions and calculations about "tensor-valued" functions, i.e. functions of the form $A : \mathbb R^d \to \mathbb R^{d^{n\times}}$.
For ...

**1**

vote

**0**answers

70 views

### Large Deviation of Triple Poisson Product

Let $X_i$ with $i=1,\ldots,n$ be independent Poisson variables, $X_i$ with parameter $\lambda_i.$
Let $\circ$ be a group operation on a group of size $n.$
I would like to obtain a large deviation ...

**6**

votes

**0**answers

176 views

### Galois action on Grothendieck ring of varieties

Let $k$ be a field and let $\overline{k}$ be its algebraic closure. Let $K_0(Var/\overline{k})$ be the Grothendieck ring of algebraic varieties over $\overline{k}$.
Is it true that the natural ...

**2**

votes

**0**answers

101 views

### Homeomorphic extension to totally disconnected sets

Dear Mathoverflow Community,
I am looking for a reference for the following topological fact:
Fact
Let $E$ and $F$ be two totally disconnected compact subsets of the plane (can assume perfect if ...

**9**

votes

**1**answer

193 views

### Reference for Schur multiplier identity

Let $G$ be a finite group and $H$ a normal subgroup of $G$. I recently stumbled upon the following identity for the Schur multiplier of $G/H$:
$$\operatorname{H}_2(G/H,\mathbb{Z}) \cong \frac{\...

**4**

votes

**1**answer

250 views

### $X(\mathbb{Z}/p\mathbb{Z})$ versus $\{X(\mathbb{Z})\pmod{p}\}$

Let $P_1$,...,$P_m$ be polynomials in $n$ variables with coefficients in $\mathbb{Z}$ and consider the set
$$X(\mathbb{Z})=\{(x_1,...,x_n)\in \mathbb{Z}^n \ |\ P_i(x_1,...,x_n)=0 ~ ,\ \forall i \in\{1,...

**4**

votes

**0**answers

96 views

### Borel selections of usco maps on metrizable compacta

The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...

**2**

votes

**0**answers

35 views

### The “semi-symmetric” algebra of a vector space

If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\...

**2**

votes

**0**answers

135 views

### Locality in Floer theory

There appears to be a dearth of resources and references for the question of 'locality' in Floer theory. In particular, I cannot seem to find any complete statement of what people refer to as '...

**7**

votes

**0**answers

94 views

### Direct sums of operator spaces

I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a ...

**3**

votes

**0**answers

176 views

### Existence and uniqueness for reaction-diffusion equations

I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$
\begin{align*}
&\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\
& u(0)=u_0\in L_2
\end{align*}
where the ...

**3**

votes

**0**answers

76 views

### Decomposability of chain complexes

The following is stated in Luc Illusie, "Frobenius and Hodge degeneration", part 4.6.
Let $L$ be a bounded chain complex. There is a sequence of obstructions, first $c_i\in \mathrm{Ext}^2(H^iL, H^{i-...

**2**

votes

**0**answers

149 views

### smooth structure on complete intersection

A complete intersection is an algebraic variety cut out by homogenous polynomials. Geometrically, this is the intersection of hypersurfaces in complex projective space.
Below, let's confine to the ...

**1**

vote

**0**answers

39 views

### GKO construction for (Super-)Virasoro algebras

I am reading the paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. In many places, the authors claim results without any justification, or with ...

**2**

votes

**0**answers

181 views

### Why is the study of homology important? [closed]

In some fields of studies, for example, Amenability of Banach algebras and $L^2$-Betti numbers, some chain complexes are studied, why is the study of these creatures important? When and why do these ...

**2**

votes

**2**answers

198 views

### $PSL_2(\mathbb{R})$ representations of free groups

Let $S_{g,n}^b$ denote a surface of genus $g$ with $n$ punctures and $b$ boundary components. Let us assume $\max\{b,n\}\geq 1$. It is then obvious that $S_{g,n}^b$ deformation retracts to a bouquet ...

**2**

votes

**0**answers

82 views

### automorphic form associated with Apollonian Gasket

In /Indra's Pearls/, it's mentioned one can associate automorphic forms with limit sets. Is there an explicit description of the one associated with the Apollonian gasket (up to some appropriate ...

**2**

votes

**0**answers

105 views

### “Base change” along étale map

Let $f \colon Y \to X$ a morphism of schemes. I denote by $f_*$ and $f^*$ the pushforward and pullback of sheaves of modules. Let $U \subseteq X$ be an open (Zariski) set, and denote by $i \colon U \...

**1**

vote

**0**answers

79 views

### Accessing older papers of Boll. U.M.I

The Bollettino dell'Unione Matematica Italiana seems to have been published by Springer since 2014, and I am unable to access any papers before that date. In fact, my university library doesn't seem ...

**5**

votes

**1**answer

97 views

### Finite posets for which all intervals are atomic

Let $P$ be a finite poset which is a lattice with $0,1 \in P$.
An atom in $P$ is an upper cover of $0$ and a coatom is a lower cover of $1$.
$P$ is atomic if every element is a join of atoms and ...

**3**

votes

**0**answers

241 views

### Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?

I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...

**4**

votes

**2**answers

106 views

### Tournament contained in vertex transitive tournament

Is it true that every finite tournament is contained in some finite vertex-transitive tournament? If not, is it known which tournaments satisfy this property? This seems like a basic question, but I ...