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There is relatively new tag straight-lines. We have previously discussed the tags line (or lines) and planes and the consensus was that the tags are not needed: Tags for lines and planes? Should the tag straight-lines be removed too?
To add more details, the tag was created a few days ago. Since...
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Let $n$ be an integer greater than $3$ and let $I(\mathbb{S}^1,\mathbb{R}^n)$ be the set of immersions from $\mathbb{S}^1$ into $\mathbb{R}^n$.
While messing around the Whitney-Grauestein theorem, I felt like the following claim is true:
Theorem. Let $f$ and $g$ be in $I(\mathbb{S}^1,\mathb...
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.
== Definition ==
Two submanifolds of a given finite-dimensional smooth manifold are said to intersect transversally if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of...
In mathematics, given a collection C of sets, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of C (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal. One variation, the one that mimics the situation when the sets are mutually disjoint, is that there is a bijection f from the transversal to C such that x is an element of f(x...
In operator theory, a discipline within mathematics, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T|| ≤ 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.
== Contractions on a Hilbert space ==
If T is a contraction acting on a Hilbert space
H
...
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M,d) is a function f from M to itself, with the property that there is some nonnegative real number
0
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1
{\displaystyle 0\leq k<1}
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≤
k
d
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{\displaystyle d(f...
