Questions tagged [partition-of-unity]
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Partition of unity without AC
Several existence theorems for partition of unity are known. For example (source),
Proposition 3.1. If $(X,\tau)$ is a paracompact topological space,
then for every open cover $\{U_i \subset X\}_{i \...
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Partition of unity of simplex
Let $$\chi_S(x,y)=\begin{cases}1&\text{ if }0< x<y< 1\\0&\text{ else }\end{cases}$$
be the indicator function of the simplex $S=\{(x,y)\in (0,1)^2:x<y\}$. I am interested in an ...
- 1,611
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The entropy of a partition of unity
A partition of unity can be thought of as a blurred out set partition: identify the fibers of the set partition with their indicator functions; the sum of these indicators is $1$. I would like to ...
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Is this matrix invertible?
Suppose that $P$ is a finite partition of the unit $d$-dimensional hypercube composed of $m$ hyperrectangles $R_1,...,R_m$
Denote $u_1,...,u_m$ the middle-point of each hyperrectangle : $u_i = \frac{...
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Partitions of unity with arbitrary Lip-constants
Lets make things simple. Suppose we have a compact metric space $(X,d)$ and then some Lipschitz partition of unity exists, say a collection $\mathcal{F}=\{f_n\}$ subordinate to some open cover $\...
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Encounters with partitions of unity
Not sure how this would be received here. This question is about smooth partitions of unity.
Let $M$ be a manifold. Consider an open cover $\{U_\alpha\}_{\alpha\in \Lambda}$ of $M$. A collection of ...
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Can we get smooth parition of unity with uniformity?
Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U_1, \ldots, U_N$, and get a smooth partition of ...
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Second Countability hypothesis for a Banach manifold
Is the second countability hypothesis necessary to rigoruosly define a Banach Manifold (say in infinite dimension)?
In the finite-dimensional theory of manifolds, that request is included in the ...
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Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?
Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.
Are there analogs of these tools for piecewise linear manifolds?
A PL ...
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Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
...
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How does the lack of partitions of unity affect the structure of analytic/holomorphic manifolds?
The standard way to define integration on a smooth manifold is to use partitions of unity, to extend to the case where the form you're integrating isn't supported on just one coordinate patch. Of ...
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symplectic form with partition on unity
Assume $M$ is a $2n-$dimensional differentiable manifold. Let $(U_{i})$ be a open covering of $M$. With respect to this covering let $\rho_{i}$ be a partition of unity. Assume that on each $U_{i}$ we ...
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What is the right version of "partitions of unity implies vanishing sheaf cohomology"
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
- 153k
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An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
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