# Questions tagged [extension]

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### Homeomorphic extension of a discrete function

Let $f : \{ 0,1 \} ^ {n} \rightarrow \{ 0,1 \} ^ {n}$ be a bijective map. Then is there a known computable way to extend it to a homeomorphism $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
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### Lie algebras for which all one-dimensional extensions split

I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will ...
139 views

### A ball with slit at the radius is not $W^{1,1}$-extension domain

Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
148 views

### Boundedness of an extension operator

Let $d \ge 2$ be a positive integer. For $x=(x_1,\dotsc,x_{d-1},x_d)$, we write $x'=(x_1,\dotsc,x_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x_d) \mid x_d>0\}$ denote the $d$-dimensional upper half-space. ...
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### Any connection between extension of algebraic structure and forcing of set theory?

Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
204 views

### Extension of first order deformations of a line bundle

Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
1 vote
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### $H^1 \cap C^0$ boundary, smooth $H^1$ extension

Assume we have a $u \in H^1(\Omega; \mathbb{R}^n) \cap C^0$ where $\Omega$ is a bounded open Set with smooth boundary. Also $u\vert_{\partial \Omega} \in H^1(\partial \Omega; \mathbb{R}^n) \cap C^0$. ...
1 vote
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### Reference for an extension theorem for Neumann boundary data

$\DeclareMathOperator\Tr{Tr}$Let $\Omega \subset \mathbb{R}^d$ be a smooth bounded domain (we denote by $n$ the normal to $\partial\Omega$) and $p\in(1,\infty)$. Do you know where I can find (book or ...
681 views

### Is there some example that nicely extends the multiplication of natural numbers?

Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The ...
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### Extension Operator for $W^{1,\infty}(U,X)$

I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
1 vote
133 views

### Computation of extension groups in the category of pairs $(M,f)$

Let $A$ be a unitary commutative ring, and let $B$ be an $A$-algebra. We consider the category whose objects are pairs $\textbf{M}=(M,f)$ where $M$ is an $A$-module and where $f$ is a $B$-linear ...
201 views

### Homomorphisms of ring extending nicely ideal intersections

Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$. Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$. Is there some "natural" assumption on $\varphi$ to ...
1 vote
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### Do all fields with internal absolute values arise as ordered fields or like $\mathbb{C}$ from them?

$\def\abs#1{\lvert#1\rvert} \def\Im{\operatorname{Im}} \def\Re{\operatorname{Re}}$ (Crossposted from math.stackexchange.com after 5 days with no correct answer.) Let ​ $\langle F,+,\cdot\rangle$ ​ be ... 2k views

### Extensions of local vector fields to whole manifold

Let $M$ be a smooth manifold (with boundary). Suppose I have a smooth vector field $T$ defined on the complement of a compact subset $K$ of $M$ and I wish to extend $T$ to the whole of $M$. What are ...
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### A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor. Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of ...
343 views

### Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$. As one could find in G.M. Troianello "Elliptic Differential Equations and ...
134 views

### Units of an extension of $\mathbb{Z}$ [closed]

Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there ...
213 views

### "Degree 3 fields"

I was wondering what was known about fields $k$ having the property that any polynomial over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ? Is there some kind ...
625 views

### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension 1\...
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### ${\rm Ext}^1$ and extensions of line bundles on a curve

I am confused about the following. I know that for two line bundles $L_1, L_2$ on an algebraic curve $C$ the vector space ${\rm Ext}^1(L_1,L_2)$ classifies isomorphism classes of rank two vector ...
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### Does every locally finite acyclic directed set embed into a linear order locally isomorphic to the integers? (Edit: extend, not merely embed.)

Let $S=(S,\prec)$ be a set together with an acyclic binary relation, generally nontransitive. $S$ is locally finite if, for every element $x\in S$, the sets $\{w|w\prec x\}$ ("direct past of $x$") ...
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### Does every countably infinite interval-finite partial order embed into the integers?

A partially ordered set $(S,\le)$ is called interval finite if the open intervals $(x,z):=\{y|x\le y\le z\}$ are finite for all choices of $x,z$ in $S$. An embedding $(S,\le)\rightarrow(S',\le')$ of ...
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### Liftability of a submodule from an associated graded module

Let $k$ be a field, $A$ a $k$-algebra (probably noncommutative), and $M$ an $A$-module that's finite-dimensional as a vector space over $k$. Let $Gr(M;k)$ denote the set of all $k$-subspaces of $M$, ...
Consider a finite-rank free $Z$-module $Y$. Let $c: Y \times Y \rightarrow Z$ be a $Z$-bilinear form. Assume that $c(y_1, y_2) + c(y_2, y_1)$ is even, for all $y_1, y_2 \in$. Then $c$ "incarnates"...
Let $H$ and $N$ be two groups with $H$ cyclic. Let $f,g:H \rightarrow \mathrm{Aut}(N)$ be homomorphisms such that $N\rtimes _f H \cong N\rtimes _g H$. Then does that mean $f(H)$ and $g(H)$ are ...