Questions tagged [embeddings]
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103
questions
1
vote
1answer
63 views
Finite graph not embeddable in $H(n,2)$
Let $n\in\mathbb{N}$ and consider $x,y \in\{0,1\}^n$. The Hamming distance of $x,y$ is defined by $$d_H(x,y) = |\{i\in \{0,\ldots, n-1\}:x_i\neq y_i\}|.$$
For $n\geq 2$ let $H(n,2)$ be the graph given ...
2
votes
1answer
125 views
Embedding CW-complexes into infinite-dimensional topological vector spaces
Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s ...
0
votes
0answers
35 views
The linear embedding complexity of subsets of $0/1$ cube
We say $\pi$ is a subset of the $0/1$ integer points in $t$ dimensions represented by coordinates $(x_1,\dots,x_t)$ of complexity $\log^ct$ if there is an $A$ of $2^{\log^ct}\times2^{\log^ct}$ ...
5
votes
1answer
190 views
When is a metric space a snowflake?
Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$.
...
4
votes
3answers
297 views
How to show that random graphs cannot be embedded with short edges
For each (not necessarily planar) embedding of a graph in $\mathbb{R}^k$ one can calculate the ratio
$$\gamma = \frac{\textsf{mean Euclidean length of edges}}{\textsf{mean Euclidean distance between ...
1
vote
0answers
56 views
Interpolation inequality involving negative Sobolev space
$\newcommand\norm[1]{\left\|#1\right\|}\newcommand\inner[2]{\langle #1,#2\rangle}$
Let $u\in \dot{H}^1(\mathbb{R}^n)$ for $n\geq 3$ where $\dot{H}^{1}$ denotes the homogeneous Sobolev space that is ...
3
votes
1answer
165 views
Is this sequence of embeddings possible?
Working in a suitable extension of $\sf Z$ like $\sf ZfC + wholeness \ axiom$, or $\sf ZFj + Reinhardt \ axiom$.
Can we have a sequence $(j_n)_{n \in \mathbb N} $ of nontrivial elementary embeddings ...
4
votes
0answers
106 views
Do holomorphic symplectic manifolds admit (high codimension) embeddings in some standard space?
Per the Whitney embedding theorem, any manifold $M$ can be embedded into a sufficiently high dimensional Euclidean space.
According to Gromov's h-principle for contact embeddings, any contact manifold ...
0
votes
0answers
85 views
Reflective embeddings outside separation?
One of the known ways to axiomatize ZF is via these two axiom schemata:
Separation: $\forall \vec{w} \forall A \exists! x: x=\{y \in A: \phi\}$
Reflection: $\forall \vec{w} \exists \alpha: \phi \to \...
6
votes
1answer
561 views
Can there exist such a sequence of elementary embeddings of the universe to itself?
Working in ZfC + Wholeness:
Can we have a countable sequence of non-trivial elementary embeddings of the universe to itself, such that the range of each embedding is a subclass of the range of its ...
23
votes
1answer
1k views
On a curious map from the complex projective plane into $S^5$
I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a ...
2
votes
1answer
181 views
Does the Hawaiian Earring Group embed into the permutation group of $\mathbb N$?
Recall that the Hawaiian earring group, $\mathbb G$, is the fundamental group of the Hawaiian Earing using the point at the origin. It can be understood more combinatorially as a subgroup of the ...
2
votes
0answers
82 views
Compact embedding of the Sobolev space $H^m(\Omega)$ and $L^2(\Omega)$ from Rellich-Kondrachov theorem
From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...
3
votes
1answer
101 views
Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation
We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...
2
votes
1answer
203 views
Fast algorithm for computing $\sum_m (n \mod m)/m!$
I'm interested in quickly computing an embedding of the profinite integers $\widehat{\mathbb{Z}}$ into the unit interval $\left[0,1\right]$.
$\widehat{\mathbb{Z}}$ can be represented as compatible ...
1
vote
0answers
158 views
Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$
Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
5
votes
0answers
88 views
Correspondence between Riemannian metrics and Euclidean embeddings
Given a sufficiently smooth manifold M,
a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely)
an embedding of M into ...
0
votes
2answers
214 views
Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?
Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$.
I would like to show that, ...
0
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0answers
102 views
Examples of (infinite) graphs which cannot be embedded into 3d space?
I was thinking about the concept of embedding graphs into Euclidean spaces. Specifically, i was looking for examples of infinite graphs which cannot be embedded in $\mathbb{R}^3$ but can be embedded ...
3
votes
1answer
108 views
Banach embedding of finite dimensional spaces
Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).
I am wondering whether are any result for the case when $r>s>2$?
...
2
votes
0answers
70 views
Dense embeddings into Euclidean space
The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
6
votes
0answers
221 views
Can a knotted sphere isometrically embed into $\mathbb R^3$?
All smooth simple closed curves in $\mathbb R^3$ (knotted or not) can be isometrically embedded into $\mathbb R^2$ as a circle of equal arclength.
The situation for knotted spheres seems more ...
5
votes
1answer
155 views
Is identification of double points of an immersion smooth?
Let $f:M^m\to N^n$ be a generic map between smooth manifolds $n>m$. Depending on the pair $(m,n)$ generic maps will have a singular set of double points $\Sigma_2\subset M$.
Let $\phi:\Sigma_2\to \...
asked Jul 3 '20 at 15:41
Warlock of Firetop Mountain
1,5431 gold badge5 silver badges17 bronze badges
1
vote
0answers
50 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
100 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
146 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
114 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
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0answers
59 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
61 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 ...
7
votes
1answer
318 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
94 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 ...
4
votes
1answer
331 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
107 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
132 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
72 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
573 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
192 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 ...
0
votes
1answer
65 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
356 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
394 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
504 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
266 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
568 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
315 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
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0answers
112 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
312 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
515 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
261 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 $\mathbb{R}^8$?
7
votes
2answers
226 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
128 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 ...
