Questions tagged [embeddings]
The embeddings tag has no usage guidance.
191 questions
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Embedding spaces and surface knots in high dimensional manifolds
This is a variation of Craig's Knot complement diffeomorphism groups and embedding spaces for a different type of very simple manifold (surfaces which have a 1-relator fundamental group instead of ...
- 1,981
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0
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34
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Realizing set systems in real space
Let $\mathcal{S} = \{S_i\}$ be a collection of subsets of the same size $s$, all drawn from the universe $[n]$, with the property that $|S_i \cap S_j| \le 1$ for all $ i \ne j$. Let us say that $\...
- 1,389
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Noncompactness of the Sobolev embedding in the critical exponent case
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$.
It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
- 446
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139
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Amalagamation of a sequence of closed immersions of schemes
Let $(X_n)_{n \geq 0}$ be a family of schemes. Let $$X_0 \to X_1 \to X_2 \to \dotsc$$ be a sequence of closed immersions (which therefore gives rise to an ind-scheme). Under which (necessarly and/or ...
- 5,482
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Intersection of weighted Sobolev spaces
Consider the Sobolev spaces with $p=2$, defined for $s \in \mathbb{R}$ as
\begin{equation}
W^{s} = \left\{ u \in \mathcal{S}', \ (1 + \lvert \cdot \rvert^2)^{{s}/{2}} \widehat{u} \in L_2 \right\}.
\...
- 2,306
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582
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Can some exotic sphere be diffeomorphically embedded into some $R^n$?
Can some (or perhaps every) exotic sphere be diffeomorphically embedded into some $R^n$? How does such an embedding (if it exists) look like? I.e., what are the equations for a particular embedding?
...
- 3,873
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Questions on J. F. Nash's answer about his errors in the proof of embedding theorem
In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked
Is it true, as rumours have it, that
you started to work on the embedding ...
- 541
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2
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220
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Normal variation of embedded surfaces [closed]
Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by
$$\phi(p,t)=\exp_p(...
- 183
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1
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182
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Do Morse functions induce embeddings?
Does the existence of Morse functions on smooth manifolds imply Whitney's embedding theorem?
(That is, given a smooth manifold $M$, does the existence of a Morse function $f:M \to \mathbb{R}$ imply ...
- 323
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0
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111
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"Nice" limits of sequences of smooth embeddings
Consider smooth embeddings of a manifold $M$ into some $\mathbb{R}^n$. If a sequence $f_k : M \to \mathbb{R}^n$ of such embeddings converges to some continuous function $f : M \to \mathbb{R}^n$, then ...
- 221
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69
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Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher
It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a $2$...
- 191
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1
answer
452
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Isotopy class of closed 2-ball embedded in R^3
My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove?
It seems like it should be easy ...
- 23
1
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1
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Nash-type theorems for Poisson manifolds
My question comes as a natural follow-up of the previous one which concerned symplectic manifolds: if $(M, P)$ is a Poisson manifold, what embedding theorems are there into some target space (I am ...
- 5,407
10
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1
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796
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Is there a Nash-type theorem for symplectic manifolds?
If $(M, \omega)$ is a symplectic manifold, is it possible to embed (or injectively immerse) it symplectically into a sufficiently large $(\Bbb R ^{2N}, \Omega)$, (with the usual symplectic structure)?
...
- 5,407
9
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embedding of quaternionic projective spaces
Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ...
- 1,990
4
votes
1
answer
199
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Besov regularity of càdlàg functions?
Let $D(\mathbb{R})$ be the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that are right continuous with left limits (also referred to as càdlàg functions). $D(\mathbb{R})$ is often called the ...
- 2,306
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1
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embeddings of product of spheres in Euclidean spaces [closed]
I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).
In general,
(1). could the product of spheres $S^{m_1}\times\cdots\times S^{...
- 2,223
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0
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222
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obstructions to embeddings of manifolds into Grassmannians
Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in $\...
- 543
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392
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Compact embedding of ${\rm L}^1_{loc}$ space
I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely:
Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle
1,2\rangle$. ...
- 165
2
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1
answer
137
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VLSI circuit embeddings
In the following paper by Valiant
http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf
He shows under theorem 2 (at the bottom of the second page) that any planar graph $G$ of degree 3 or 4 ...
- 903
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340
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Embeddings between weighted Besov spaces
Consider the Besov spaces $B_{p,q}^s(\mathbb{R}^d)$ for parameters $0<p,q\leq \infty$ and $s\in \mathbb{R}$. The weighted Besov space $B_{p,q}^s(\mathbb{R}^d;\mu)$ is defined for $\mu \in \mathbb{R}...
- 2,306
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254
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Equivariant isometric embedding of manifolds in a Hilbert space under a noncompact group action
Given a Riemannian manifold $M$ and a group of isometries $G$ of $M$, I am interested in when there exists a isometric embedding $\iota : M \to H$, where $H$ is a Hilbert space and a representation $\...
- 2,834
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593
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When is the identity Hilbert-Schmidt between weighted Sobolev spaces?
Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space
$$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := \...
- 2,306
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1
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700
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Embedding graphs into hyperbolic spaces
Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?)
I don't ...
- 617
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2
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374
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Transformations that leave the Plucker embedding of G(2,4) invariant
I am interested in a group of transformations that leave the Plucker embedding of complex Grassmannian $G(2,4)$ into $CP^5$ given by $\lambda_{12}\lambda_{34}-\lambda_{13}\lambda_{24}+\lambda_{14}\...
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277
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Complements of unknotted tori (higher dimensions)
It is well-known that an unknotted 2-torus in $S^3$ provides the standard Heegaard splitting, in particular its complement consists of two solid tori.
It is also known that an unknotted 3-torus in $S^...
- 10.4k
1
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1
answer
621
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Condition to obtain a not compact embedding
I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ ...
- 277
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Global geometry measures for Riemannian manifolds
I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
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3
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618
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Mathematical value of constructing sphere eversions
I am extremely impressed by the work that has been done constructing sphere eversions, and other similar explicit geometrical proofs. In particular, surely nobody can fail to be impressed by the ...
- 2,513
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Embedding Euclidean buildings into products of trees
A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.)
Question: Is it true ...
- 83
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Are there non-trivial graphs that uniquely embed to hypercubes?
The $n$-dimensional hypercube is the graph formed by $0$-$1$ sequences of length $n$ where two vertices are adjacent if they differ at only one place.
The weight of a sequence is the number of $1$'s ...
- 19.1k
3
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1
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Polynomial (non-)embedding of a simplex in euclidean space
Let $\Delta$ be a standard $k$-simplex, and let $f:\Delta\to\mathbb R^N$ be a polynomial map with known numerical coefficients. What sort of practical computational algorithms can be used to ...
- 301
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2
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379
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Number of edges in linklessly embeddable graphs
What is the maximum number of edges of an $n$-vertex linklessly embeddable graph?
A more general question is the following. What is the maximum number of edges of an $n$-vertex graph with Colin de ...
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3
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Contractibility of space of embeddings of a disc
I'm pretty sure that both of the following spaces are contractible. However, I can't seem to find a proof or a reference. Can anyone provide one? Let $D^2$ be the unit disc in $\mathbb{R}^2$.
The ...
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139
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Disks in Flat Embeddings of Graphs in $\mathbb{R}^3$
Robertson, Seymour and Thomas proved that any linkless graph $G$ has a flat embedding in $\mathbb{R}^3$ (see for example A survey of linkless embeddings). An embedding of $G$ is flat if for any cycle $...
- 415
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Different definitions of linkless graphs
Robertson, Seymour and Thomas defined linkless embeddings of graphs as follows:
An embedding of $G$ is linkless if every pair of disjoint circuits of $G$ have zero linking number (see here). However ...
- 415
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1
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452
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Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space?
Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space?
Moreover I would like to know if any ...
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Is a Morse function always the height function of some embedding? [closed]
Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of a torus into $\mathbb{R}^3$ is drawn, and the Morse ...
- 5,565
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Homotopy groups of spaces of embeddings
Let $\mathrm{Emb}(M, N)$ be the space of smooth embeddings of a closed manifold $M$ into a manifold $N$ equipped with smooth compact-open topology.
Question 1. Are there conditions ensuring that ...
- 29.1k
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Monotone embedding of complete binary tree in hypercube
Embedding different graphs, especially binary trees, in the hypercube has a huge literature. However, I could not find anything if we restrict the embedding to be monotone. So I would like to ...
- 19.1k
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When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
