Questions tagged [embeddings]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
0answers
44 views

What is known about this generalization of planar dual?

So it is well known that given a planar graph, $G$, embedded in the plane (without edge crossing, so a planar embedding). One can construct the planar dual, $G^*$. What is perhaps slightly less well-...
3
votes
0answers
95 views

Which metric spaces embed isometrically in $\ell_p$?

It is known that each metric space $X$ embeds isometrically in the Banach space $\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
2
votes
1answer
59 views

Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension?

Background: I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded ...
2
votes
1answer
101 views

complemented subspace of the direct sum of two Banach spaces

When I was reading a paper, I saw something like: If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented ...
5
votes
0answers
51 views

subanalytic realization of smooth abstract stratification

Consider an $C^\infty$ abstract stratification $A$ (in the Thom-Mather sense, see Mather's note). Can we embed $A$ in some $\mathbb{R}^n$ (or in an analytic manifold) as a subanalytic set? If not, ...
3
votes
0answers
50 views

Embedding abelian categories into abelian sheaves

The Yoneda functor from an abelian category into sheaves of abelian groups is shown to be exact in The Stacks Project, Lemma 19.9.2. I like this proof because it is constructive and it doesn't use ...
6
votes
1answer
182 views

An orientable surface that cannot be embedded into $\Bbb R^3$? [duplicate]

I previously asked this question on MSE, without success. By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$. Now, Wikipedia states in this ...
2
votes
0answers
87 views

Open embedding of non-separable infinite dimensional manifolds

It is well-known (see here) that separable infinite-dimensional topological Hilbert manifolds can be embedded as open sets of the modeling separable Hilbert space. Using that separable Fréchet (in ...
6
votes
1answer
294 views

Which topological spaces admit embeddings into Euclidean spaces

I'm interested in the dual question to: continuous images of open intervals, about surjections onto open intervals. Namely, if $X$ is a topological space, when can we guarantee that there exists a ...
2
votes
0answers
95 views

Embedding a binary subspace to $l_2$ in a much lower dimension

I'm trying to find a way to embed a binary linear subspace of dimension $n$ (a linear code) to the Euclidian space while reducing the dimension significantly. The subspace (or code) contains points ...
2
votes
1answer
119 views

subelliptic Sobolev compact embedding theorem

Consider the smooth vector fields $X=(X_1,X_2,...,X_m)$ defined in a open bounded set $\Omega\in R^n$. And the non-isotropic dimension is $Q$ (see https://arxiv.org/pdf/1502.06332.pdf page 398) In the ...
3
votes
0answers
56 views

Embedding a continuous-time martingale in Brownian motion

Using the Skorohod embedding, we can embed any square-integrable discrete time martingale $(M_n)$ into a Brownian motion, obtaining times $(T_n)$ such that $(B(T_n))_{n\ge 0}$ is a version of $(M_n)$. ...
9
votes
1answer
439 views

Normal bundle of Whitney embedding

Let $X$ be a real $n$ dimensional manifold. One knows that it can be embedded into $\mathbb{R}^{2n}$ by the Whitney embedding theorem. The normal bundle for such an embedding will be a rank $n$ real ...
4
votes
1answer
165 views

Generalization of Gagliardo-Nirenberg Inequality

The standard Gagliardo-Nirenberg Inequality is $$ \Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le C_n \Vert \nabla u\Vert_{L^{1}(\mathbb R^n)}, \tag{$\ast$}$$ and constitutes a key step to proving ...
2
votes
1answer
59 views

Lower Estimate of A Lipschitz Map

Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function $\rho:(0,\infty)\...
3
votes
2answers
300 views

Is Cohen immersion conjecture (theorem) known for vector bundles?

R. Cohen proved the immersion conjecture in a 1985 Annals paper: Cohen, Ralph L., The immersion conjecture for differentiable manifolds, Ann. Math. (2) 122, 237-328 (1985). ZBL0592.57022. Any smooth ...
3
votes
1answer
242 views

Local diffeomorphism on a neighborhood of an embedding

In my reading of the (excellent!) paper of Grabowski and Rotkiewicz on higher vector bundles (https://arxiv.org/abs/math/0702772), I have encountered the following argument which I do not understand. ...
5
votes
2answers
435 views

Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is ...
1
vote
1answer
202 views

Hyperbolic embedding of a directed acyclic graph defined over strings

For integer $n$ and alphabet $\Sigma$ we construct a DAG (directed acyclic graph) $G=(V,E)$ over strings $s\in\Sigma^\star$ as follows: $$V = \{s\in\Sigma^\star\colon |s|\le n\}$$ $$E = \{(s_1,s_2)\...
7
votes
2answers
441 views

Embeddings of flag manifolds

Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...
7
votes
2answers
308 views

About the commutativity of the $1^{st}$ homotopy group of the space of knots

I would like to know if the fundamental group of the connected component of a knot space could be non commutative. I am specially interested in the case of $\mathbb{R}^3$, $\mathbb{S^3}$ or some other ...
0
votes
0answers
101 views

Intrinsic Reach for a Riemannian manifold

The reach of a set $X\subseteq \mathbb{R}^d$ is the supremum of all $r \geq 0$ such that for all $y\in X^c$ with $dist(y,X)<r$ there is a unique $x\in X$ with $dist(y,x)= dist(y,X)$. My question: ...
-3
votes
1answer
299 views

Can there be elementary embedding between a universe and a universe inside it?

[EDIT] the prior question (see the second section below) was trivially false, however the intention is to arrange a possible world of such universes, in other words the question is about if it is ...
9
votes
1answer
485 views

A wild embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$

Can one construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$ so that every orthogonal projection onto a two dimensional plane is a unit disc? It is easy to construct an embedding of $\...
2
votes
0answers
247 views

Embedding of $CP^2/CP^1$ into euclidean space [closed]

Is there a "nice" embedding of $\mathbb{C}\mathbb{P}^2/\,\mathbb{C}\mathbb{P}^1$ into $\mathbb{R}^8$?
6
votes
2answers
187 views

Embedding open connected Riemann Surfaces in $\mathbb{C}^2$

This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
0
votes
1answer
123 views

$f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$?

Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we ...
12
votes
1answer
428 views

Embedding Riemannian manifolds into some infinite dimensional manifolds?

First of all I am new to the field of embedding one manifold into another other. I have recently come across with the paper "Embedding Riemannian manifolds by their heat kernel" by P. BERARD, G. ...
5
votes
0answers
92 views

Theoretical justification of time-series forecasting using Takens' embedding

This is a cross-posting where I couldn't get an answer. In the meantime I have tried to improve the original logic: As in Takens original paper about his embedding theorem, consider a compact $m$-...
1
vote
1answer
135 views

Trouble with plane embedding

Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane. Are there any general results which can be ...
1
vote
0answers
121 views

Max-min genus of a bipartite graph

As usual, the genus of a graph with a prescribed circular ordering of the edges at each vertex is defined as the minimum genus of an orientable surface in which the graph can be drawn without edge ...
3
votes
1answer
277 views

Generalizations of Abhyankar-Moh theorem (embeddings of the line in the plane)

Abhyankar-Moh theorem says that if $L$ is a complex line in the complex affine plane $\mathbb{C}^2$, then every embedding of $L$ into $\mathbb{C}^2$ extends to an automorphism of the plane. It seems ...
13
votes
3answers
1k views

When does homology represent an embedded sphere?

If we have a triangulation of a manifold $M$ of dimension $i$ and we have simplicial homology $H_i(M)=\mathbb{Z}$, what is the condition than there exists an embedded sphere $S^i$ that generates the ...
4
votes
1answer
411 views

Is the space of tangents actually the tangent space?

This is a crosspost of this MSE question. Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}_pX$ denote ...
11
votes
1answer
232 views

Can $\Delta_1$ (or $\Delta_0$)-elementary embeddings from $V$ to $V$ exist?

Suppose V is a model of Godel-Berney's set theory with the axiom of choice. A well-known result of Kunen says that there can be no elementary embedding $V$ to itself. This result further implies that ...
2
votes
1answer
489 views

Second fundamental form and embeddings

Let $\Sigma$ be a smooth hypersurface of a $d$ dimensional smooth Riemannian manifold $(\mathcal M, G)$; we may see $G_x$ as a mapping from $T_x(\mathcal M)$ into $T_x^*(\mathcal M)$ so that $$ \...
1
vote
1answer
100 views

On graph imbedding genus clarification

Given a graph the minimum genus $g$ is the minimum number of handles needed so that there an imbedding of the graph on the surface with no edge crossings. If the graph is of genus $g$ then is there ...
4
votes
2answers
165 views

Number of non-equivalent graph embeddings

Given a graph $G$, there is a minimal integer $g$ associated with it which captures the minimum genus a surface needs to have so that $G$ embeds in the surface without edge crossings. Is there a way ...
8
votes
0answers
82 views

Is there a quaternionic analogue of Kodaira's embedding theorem?

Let $M$ be a $4m$-dimensional Quaternion-Kähler manifold of positive scalar curvature. Does there exist an $n$ large enough, so that $M$ can be embedded inside $\mathbb{H}P^n$ via a quaternionic ...
4
votes
0answers
165 views

Whitney-like embedding theorem for posets?

The Whitney embedding theorem says that any finite-dimensional smooth manifold can be embedded into $\mathbb{R}^n$ for some $n$. Is anything like this true for posets? I'm looking for conditions on a ...
4
votes
1answer
168 views

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
vote
0answers
32 views

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 $\...
5
votes
0answers
406 views

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 ...
4
votes
0answers
135 views

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 ...
0
votes
0answers
141 views

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\}. \...
0
votes
0answers
484 views

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? ...
23
votes
2answers
3k views

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 ...
0
votes
2answers
152 views

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(...
3
votes
1answer
138 views

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 ...
1
vote
0answers
101 views

“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 ...