Questions tagged [embeddings]

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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 ...
NullOfMatrix's user avatar
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0 answers
109 views

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)....
Yromed's user avatar
  • 63
1 vote
0 answers
121 views

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 $\...
M. Winter's user avatar
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7 votes
1 answer
243 views

Does the continuous image of a disc contain an embedded disc?

Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
M. Winter's user avatar
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1 vote
0 answers
82 views

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)}...
user515999's user avatar
3 votes
0 answers
111 views

Is it consistent to have an infinite antitone sequence of elementary embeddings such that the involved models include iterated sharps?

$\DeclareMathOperator\crit{crit}$Background essays (the material I've tried to understand in leading up to this question): Daghighi, et. al. [2014], "The foundation axiom and elementary self-...
Kristian Berry's user avatar
0 votes
1 answer
93 views

Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$?

For any $\kappa>0$, we consider the Gaussian heat kernel $$ p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}}, \quad t>0, x \in {\mathbb R}^d. $$ Let $L^0 := L^0 (\...
Akira's user avatar
  • 749
6 votes
1 answer
315 views

Different flavours of Vassiliev Conjecture

There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
Andrea Marino's user avatar
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43 views

Is a RKHS defined using a feature map over another RKHS bigger than the latter RKHS?

I am interested in learning more about what happens when 'composing' two reproducing kernel Hilbert spaces (RKHS). Let $\phi\in C(\mathbb{R})$ and $X=[-1,1]^d$. Suppose we have two RKHSs with the ...
ChocolateRain's user avatar
4 votes
0 answers
218 views

stating large cardinal axioms in ZF

Can I ask whether there is a good reference for how to state the standard large cardinal axioms in the context of $ZF$? My concern is that it seems that the usual proof that embeddings defined from ...
Rupert's user avatar
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1 vote
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118 views

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 ...
Justin_other_PhD's user avatar
3 votes
0 answers
68 views

Is every weakly $1$-dimensional space embeddable in the plane?

A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$ is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
D.S. Lipham's user avatar
  • 3,045
4 votes
2 answers
335 views

Knot theory in handlebodies of arbitrary genus

It is well known that not all graphs embed on the plane (e.g. the graph $K_{3,3}$). However, every finite graph embeds into a surface of some genus. One can think of this procedure as starting with a ...
João Lobo Fernandes's user avatar
6 votes
0 answers
144 views

Metric spaces containing a topological disc

It is well-known that every connected, locally connected compact metrizable space $X$ contains an arc, that is, a subspace homeomorphic to $[0,1]$. Are there topological properties we can add to these ...
Jeremy Brazas's user avatar
4 votes
0 answers
58 views

Which cellular embeddings of Eulerian graphs have bipartite duals?

It is well-known that a plane graph $G$ is Eulerian if and only if its (geometric) dual $G^*$ is bipartite. I am interested in generalisations of this result to cellular embeddings of Eulerian graphs ...
Cyriac Antony's user avatar
5 votes
3 answers
530 views

If a polyhedron in $\mathbb{R}^3$ has local intersections, does it also have more global intersections?

Consider a simplicial complex $K$. A piecewise linear map $f: K \to \mathbb{R}^n$ is an almost-embedding if $f(\sigma) \cap f(\tau) = \emptyset$ for any two disjoint simplices $\sigma,\tau$ in $K$. ...
Omega Tree's user avatar
2 votes
1 answer
206 views

Sufficient condition for the union of two submanifolds to be a submanifold

I have two smoothly embedded orientable surfaces $S_1,S_2\subset S^3 \times [0,1]$ with boundary such that $(i)$ $S_1\cap S_2$ is a smoothly embedded surface without boundary and $(ii)$ $\overline{...
Euler Characteristic's user avatar
4 votes
0 answers
138 views

Applications of Strong Whitney Embedding

I am looking for applications of the strong Whitney's embedding theorem that have an advantage over weak theorems. That is, applications where it's important that the dimension of the Euclidian space ...
Ludwik's user avatar
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4 votes
0 answers
259 views

Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?

Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$. Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...
M. Winter's user avatar
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2 votes
0 answers
114 views

How to express Kunen's inconsistency, Reinhardt and Wholeness axioms, by single sentences?

Working in $\sf NBG, $ can we express the property of a class being set theoretically definable, by a single sentence? Like for example, the following way: $$\operatorname {std}(X) \iff \exists x_1 \...
Zuhair Al-Johar's user avatar
-3 votes
1 answer
66 views

Sobolev embedding [closed]

I was trying to understand Sobolev embedding, some results about this topic are not clear to me. My question is the following: what are the condition on $p_1 , \alpha_1, p_2 $and $\alpha_2$ for $W^{...
Said Kamam's user avatar
4 votes
0 answers
242 views

Can we have full choice prior to Reinhardt cardinals?

Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
Zuhair Al-Johar's user avatar
1 vote
0 answers
65 views

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 ...
Zuhair Al-Johar's user avatar
2 votes
1 answer
329 views

Can we interpret Reinhardt cardinals this way?

To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the ...
Zuhair Al-Johar's user avatar
1 vote
2 answers
258 views

Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?

Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves? The following is a formal capture of that idea: To the language of $\sf ZF$...
Zuhair Al-Johar's user avatar
8 votes
1 answer
1k views

Is there a form of choice that can elude Kunen's inconsistency theorem?

When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, ...
Zuhair Al-Johar's user avatar
0 votes
0 answers
104 views

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{...
Andrea Marino's user avatar
0 votes
1 answer
86 views

Hadamard submanifolds of $k$-fold product of hyperbolic plane

Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian ...
Math_Newbie's user avatar
6 votes
0 answers
145 views

What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?

Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...
Random's user avatar
  • 885
3 votes
1 answer
268 views

If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength?

If we add to the wholeness axiom, the axiom of stratified\acyclic replacement, what would be the consistency state and strength of the resulting theory? The wholeness axiom $\sf WA$, introduced by ...
Zuhair Al-Johar's user avatar
7 votes
1 answer
322 views

Can we have mutual elementary embeddability between distinct transitive sets?

Is it consistent with $\sf ZFC$ to have mutual elementary embeddability between distinct transitive sets? Formally, is the following theory consistent? $${\sf ZFC} + \exists M \exists N: M,N \text { ...
Zuhair Al-Johar's user avatar
5 votes
1 answer
193 views

How to formalize this isotopy?

I'm studying the H-Cobordism theorem following the Lectures of John Milnor, and in the proof of the Whitney trick for cancel pairs of self-intersection points I have the next problem with an isotopy ...
Ludwik's user avatar
  • 237
4 votes
0 answers
97 views

Mostow-Palais equivariant embedding for manifolds with corners

Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into ...
Laurent Cote's user avatar
4 votes
0 answers
246 views

Explicit formula for embedding of real projective spaces into Euclidean spaces

I am interested in representing vectors in $\mathbb{R}^n$ in a sign-invariant and efficient manner. That is, I am looking for a function $$f:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^d$$ such that for $v\...
Felix Crazzolara's user avatar
5 votes
1 answer
228 views

Space of spacelike embeddings as infinite-dimensional manifold

Consider a four-dimensional Lorentzian manifold $(\mathcal{M},g)$ and a $3$-dimensional compact manifold $\Sigma$, such that there exists a spacelike embedding $i:\Sigma\to\mathcal{M}$ so that $h:=i^{\...
G. Blaickner's user avatar
  • 1,137
3 votes
1 answer
207 views

Adding a self-intersection point to an immersion

I'm currently working on strong Whitney's embedding theorem, using Adachi's notes on Embeddings and Immersions. But I am stuck on a statement that Adachi comments without proof about adding a unique ...
Ludwik's user avatar
  • 237
1 vote
0 answers
115 views

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 $1 \leqslant i \leqslant [n/2]$. ...
user488802's user avatar
8 votes
1 answer
372 views

Whitney embedding theorem for Hölder manifolds

According to a result of Whitney any $C^r$-manifold, $r\geq 1$, is $C^r$-homeomorphic to a smooth embedded submanifold of some Euclidean space; see Theorem 1 in Hassler Whitney, "Differentiable ...
Christian Lange's user avatar
1 vote
0 answers
101 views

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,...
KKD's user avatar
  • 463
5 votes
1 answer
840 views

Can a smooth manifold be realised as the image of a smooth function?

Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$? $X$ may have points at which the ...
dennis's user avatar
  • 423
1 vote
2 answers
217 views

Ricci scalar of submanifold of $\mathbf R^n$

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} where $\vec x$ are ...
dennis's user avatar
  • 423
2 votes
1 answer
268 views

Volume of submanifold as integral of delta-function

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} (where $\vec x$ are ...
dennis's user avatar
  • 423
3 votes
0 answers
177 views

Volume of sub-manifold of $\mathbf R^n$

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations: \begin{equation} P_1(\vec x)=0, \\ \vdots \\ P_m(\vec x)=0, \end{equation} (where $\...
dennis's user avatar
  • 423
3 votes
1 answer
179 views

Topologically embed Klein bottle into $\mathbb{R}^4$ projecting to usual "beer-bottle" surface in $\mathbb{R}^3$

(Originally asked in 2018 at https://math.stackexchange.com/questions/2946505/topological-embedding-of-klein-bottle-into-mathbbr4-that-projects-to-usual?noredirect=1#comment9514257_2946505;cross-...
murray's user avatar
  • 153
4 votes
1 answer
207 views

Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?

Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$. ...
Somatic Custard's user avatar
3 votes
0 answers
152 views

Constant in Naor and Neiman's Assouad Theorem

In Naor and Neiman's Assouad embedding theorem - "Assouad’s theorem with dimension independent of the snowflaking" Revisita Mathematica, the authors derive quantitative estimates on the ...
ABIM's user avatar
  • 4,989
3 votes
0 answers
87 views

Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?

Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
M. Winter's user avatar
  • 12.5k
1 vote
0 answers
135 views

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 $...
user488802's user avatar
4 votes
0 answers
166 views

In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?

I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...
M. Winter's user avatar
  • 12.5k
3 votes
1 answer
116 views

Holomorphic/Symplectic embedding of Riemann surfaces

Let $\Sigma_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma_g \times \Sigma_g$. Then can there exist holomorphic embeddings of $\Sigma_l$ into $X$ for $l < g$? What about ...
cr1t1cal's user avatar
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