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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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 ...
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1answer
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. ...
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0answers
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 \...
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2answers
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 ...
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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
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1answer
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 ...
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1answer
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
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1answer
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 ...
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2answers
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 $...
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3answers
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 ...
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0answers
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 ...
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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 ...
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5answers
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 ...
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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 ...
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2answers
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 ...
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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 ...
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1answer
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}...
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0answers
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 $R$ ...
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3answers
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
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0answers
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 ...
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0answers
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 ...
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0answers
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: ...
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7answers
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 ...
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1answer
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-...
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0answers
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 $...
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2answers
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
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0answers
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
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1answer
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} \...
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0answers
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 ...
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0answers
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 ...
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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
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0answers
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
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1answer
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
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1answer
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
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0answers
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
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0answers
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
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0answers
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 '...
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0answers
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
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0answers
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 ...
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0answers
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-...
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0answers
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 ...
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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 ...
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0answers
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
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2answers
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
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0answers
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
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0answers
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 \...
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0answers
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
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1answer
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
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0answers
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
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2answers
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 ...