# Questions tagged [embeddings]

The embeddings tag has no usage guidance.

80
questions

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### 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-...

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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 ...

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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 ...

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votes

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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 ...

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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, ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $(M_n)$. ...

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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 ...

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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 ...

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**1**answer

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)\...

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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 ...

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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. ...

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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 ...

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vote

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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)\...

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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 ...

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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 ...

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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: ...

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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 ...

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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 $\...

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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 $\mathbb{R}^8$?

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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 ...

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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 ...

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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. ...

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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 $m$-...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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
$$
\...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $\...

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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 ...

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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 ...

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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\}.
\...

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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?
...

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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 ...

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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(...

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**1**answer

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 ...

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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 ...