Skip to main content

# Questions tagged [extension]

The tag has no usage guidance.

58 questions
Filter by
Sorted by
Tagged with
0 votes
0 answers
36 views

### (optimal) Constant in Sobolev extension Theorems

It is well known that for Lipschitz domains $\Omega\subset \Bbb{R}^2$ one can define linear extension operators $$E: H^1(\Omega) \to H^1(\Bbb{R}^2).$$ I am interested in explicit upper bounds for the ...
• 2,172
1 vote
1 answer
96 views

• 143
2 votes
0 answers
137 views

### Homotopy equivalence of chain complexes from subcomplexes and quotient complexes

Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
• 653
5 votes
1 answer
174 views

### On the property P in the Whitney extension theorem

Let $D$ be a possibly unbounded domain in $\mathbb{R}^d$, $d \ge 2.$ We say that $D$ has the property P if there exists $C>0$ such that such that any pair of points $x,y \in D$ can be joined by a ...
• 701
1 vote
0 answers
36 views

2 votes
1 answer
408 views

• 181
4 votes
0 answers
145 views

### Continuous extension preserving modulus of continuity

Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
2 votes
0 answers
263 views

### Extension of Lipschitz functions that preserve the Frobenius norm of the Jacobian

Let $n,m\ge 1$ be integers and let $f:E\to R^m$ be $L$-Lipschitz for some subset $E\subset R^n$. Kirszbraun's theorem, https://en.wikipedia.org/wiki/Kirszbraun_theorem, states that there exists ...
• 1,564
2 votes
0 answers
56 views

### Extension of differentiable structure to guarantee continuous extension of prescribed vector fields

Consider the lower half space $\{(x,y) \in \mathbb{R}^2 \; | \; y < 0\}$ and let $X_1, X_2$ be two vector fields on the lower half space which are continuous with respect to its canonical ...
• 383
6 votes
0 answers
234 views

### How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
1 vote
0 answers
48 views

• 1,623
21 votes
1 answer
752 views

### Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure

Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-...
• 12.8k
1 vote
0 answers
154 views

• 351
3 votes
1 answer
409 views

### busby invariant of extensions of $C^*$-algebras

I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras". Let $0\to B\to E\to A\to 0$ be a short exact ...
1 vote
0 answers
448 views

### Quotient of two smooth functions extension

Assume we are given smooth functions $f, g: U \to \mathbb{C}$, where $0 \in U \subset \mathbb{R}^n$ is open and $0 \in g^{-1}(0) \subset \{x_n = 0\}$. Furthermore, suppose that $\nabla g \neq 0$ on ...
• 501
5 votes
1 answer
350 views

### Extension of functions from geodesically convex compact sets in a Riemannian manifold

In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
• 34.8k
9 votes
0 answers
113 views

### General approaches to extension theorems as Caratheodory

I would like to know if there are some general studies about extension-like theorem, in the sense which i'm going to describe. This paragraph is not rigorous; I just would like the idea to be clear. I ...
3 votes
1 answer
3k views

### Extension of continuous and smooth functions

Let us consider any subset $U \subset \mathbb{R}^{n}$. By definition, a function $f: U \rightarrow \mathbb{R}^m$ is smooth if, for every $x \in U$, there exist an open neighbourhood $\Omega_{x}$ of $x$...
• 1,222
3 votes
1 answer
869 views

### Fractional Sobolev spaces and extension by zero

The all-wise Wikipedia suggest (https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero) that for $s > 1/2$ any function from $H^s_0(\Omega)$ can be extended by zero to whole $\mathbb{R}^n$ (...
5 votes
2 answers
453 views

### Conjugation in associative algebras over finite fields

Let $A$ be a finite dimensional associative algebra (with unity) over a finite field $F$. Let $L$ be a field extension of $F$. Suppose that after extending scalars to $L$, two elements $a,b$ of $A$ ...
• 367
3 votes
1 answer
2k views

### Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?

A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...
• 139
-1 votes
1 answer
207 views

### Ext functor for more than two modules? [closed]

The question is natural. Let's just work in the category of modules over a ring. Pick three modules $M_1, M_2, M_3$. Consider consecutive extensions of these modules, i.e., consider M, such that we ...
• 11
4 votes
1 answer
305 views

### Triviality of local system extension

Take a nice space $X$. Let us call a local system on $X$ a functor from the fundamental groupoid of $X$ to groups, so that $G$ is a local system on $X$ if for each $x \in X$ there is a group $G_x$ and ...
17 votes
2 answers
2k views

• 1,871