Opened 16 months ago
Last modified 5 months ago
#31707 new enhancement
Manifold of piecewise linear functions with k marked breakpoints
Reported by: | mkoeppe | Owned by: | |
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Priority: | major | Milestone: | sage-9.7 |
Component: | manifolds | Keywords: | |
Cc: | yzh, egourgoulhon, gh-mjungmath, tscrim | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description
We define the Riemannian manifold PL_k of continuous piecewise linear functions from the interval [0,1] to the reals with k+1 marked breakpoints.
Default chart: breakpoints 0 = a_0 < a_1 < a_2 < ... < a_k = 1; slopes s_1, ..., s_k.
The breakpoints are marked: For example for k = 2, we distinguish the constant function with a_0 = 0, a_1 = 1/2, a_2 = 1 and s_1 = s_2 = 0 from the constant function with a_1 = 1/3.
This manifold has an immersion (but not embedding) into the Hilbert space of L^{2} functions. The inner product there (see #30218) pulls back to define the metric on PL_k.
Elements of PL_k indicate their embedding.
Change History (4)
comment:1 Changed 16 months ago by
- Cc tscrim added
comment:2 Changed 13 months ago by
- Milestone changed from sage-9.4 to sage-9.5
comment:3 Changed 8 months ago by
- Milestone changed from sage-9.5 to sage-9.6
comment:4 Changed 5 months ago by
- Milestone changed from sage-9.6 to sage-9.7
Short of defining infinite-dimensional Banach and Hilbert manifolds, for this ticket we would define
ContinuousMap
that maps into an arbitrary (topological)VectorSpace
(with distinguished basis)DiffMap
that maps into an arbitrary (topological)VectorSpace
orInnerProductSpace
(#30218) (with distinguished basis)TopologicalSubmanifold
,DifferentiableSubmanifold
,PseudoRiemannianSubmanifold
that can work with these types of maps.