Questions tagged [embeddings]
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82 questions with no upvoted or accepted answers
2
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131
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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 ...
- 301
2
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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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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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51
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Compact embeddings RKHSs into Sobolev Spaces
Let $\mathcal{H}$ be an RKHS over an open domain $\Omega \subseteq \mathbb{R}^d$. Are there conditions under which $\mathcal{H}$ can be compactly embedded in a Sobolev space $W^{s,p}(\Omega)$ for ...
1
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0
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75
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Closed regular monomorphism and closed subspace
I have read the categorical definition of embedding, which is that of regular monomorphism. In the case of the category of locally ringed space, is it true that closed subspaces are defined by closed ...
- 35
1
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77
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Is there a variant of the crossing lemma for multigraphs with arbitrary embedding?
Suppose $G$ is a graph embedded in the plane with $m=|E(G)|$ edges and $n=|V(G)|$ vertices.
Suppose $\operatorname{sim}(G)$, the simplification of $G$ contains $ m' \gg 3n $ edges.
Call the set of ...
- 111
1
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0
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141
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Can a closed null-homotopic curve be filled in by a disc?
Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...
- 13.6k
1
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123
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Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
1
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0
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125
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Do cycle graphs embed isometrically in spheres?
I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track ...
- 169
1
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79
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Can this method let choiceless large cardinals be smaller than cardinals compatible with choice?
Recall question "Can we have this sequence where choice fails and returns?"
Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension ...
- 11.3k
1
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119
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Embedding (Kronecker product) preserves the structure?
In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix}
-I_{i} & 0\\
0 & I_{n-i}
\end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
- 501
1
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0
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118
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Exists $G$-equivariant embedding with faithful representation of $G$?
Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...
- 473
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136
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Realization of a subgroup in a maximal subgroup of a classical group
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$In the finite group $G = \operatorname{PGL}_8(5)$, there is an elementary abelian $2$-subgroup $E$ of rank 5. $E = A_{1} \times A_{2} $ where $...
- 501
1
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0
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106
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Best estimate on doubling constant of a finite metric space
Let $(X,d)$ be a finite metric space. Clearly, $(X,d)$ is a doubling metric space but is there a 'best' estimate of $(X,d)$'s doubling constant?
Probability based on its cardinality, diameter, and ...
- 195
1
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31
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Separability of graph component embeddings
I have an undirected graph. I also have an embedding of the graph in $\mathbb{R}^3$. Assume that the graph has 2 connected components. I want to know whether there exists a plane (or better - any ...
- 283
1
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0
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112
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The best lower bound for isometric immersions
I just read Azov's article in the considered two classes of Riemannian metrics,
\begin{align*}
ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\
ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2
,&g>0\...
- 143
1
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70
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Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
1
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109
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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-...
- 461
1
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212
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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: ...
- 3,574
1
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0
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159
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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 ...
- 1,389
1
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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
1
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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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67
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Constants in the entropy number of the Sobolev space
For a Sobolev space with $W^s(\Omega)$, where $\Omega\subset R^d$ is a compact space with smooth boundary, we know that the entropy number satisfies $e(\delta, W^s(\Omega, 1),\|\cdot\|_{L_\infty})\leq ...
- 39
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154
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Embedding of the first Hirzebruch surface in $\mathbb{P}^4$ as a cubic surface
The first Hirzebruch surface (the blow-up of $\mathbb{P}^2$ at one point) is a projective toric surface that naturally embeds into $\mathbb{P}^4$ as a cubic surface (sometimes called the cubic scroll)....
- 183
0
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111
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Name for homotopy totalization of Goodwillie tower (in embedding calculus)
Let $M,N$ be a manifold and consider the presheaf of spaces $\textrm{Emb}(-, N)$ on the open sets of $M$. Classical embedding calculus produces a goodwillie tower
$$ \ldots \rightarrow T_{k+1} \textrm{...
- 2,152
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155
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Prove that Takens' embedding is a smooth one-to-one map with a smooth inverse
Let $f: \mathcal{M} \rightarrow \mathcal{M}$ be a smooth diffeomorphism and $\phi: \mathcal{M} \rightarrow \mathbb{R}$ be a smooth function, where $\mathcal{M}$ is a $d$-dimensional manifold (which we ...
- 297
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241
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Can a non-reflexive space embed into a reflexive space?
My question is inspired from the concept of super-reflexivity which was defined by James here: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/superreflexive-banach-...
- 11
0
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0
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162
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Compact embedding of anisotropic Sobolev space
I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
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94
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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 \...
- 11.3k
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377
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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 ...
- 11
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216
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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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581
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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