Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Characteristic crossing and caustic for a first-order PDE
Definition
For a characteristic strip, a crossing or caustic is a point or time at which the projected map loses local rank or local one-to-one graphing. It is a failure of the projection needed to define , not a claim that the lifted ODE has reached its maximal lifespan. For a forward-time family, a first crossing time is an infimum in , with value when the relevant set is empty.
Depends on
Used by
- Quasilinear characteristics can cross before the lifted ODE blows up Counterexample
- Quadratic Hamilton–Jacobi data produce an explicit caustic time Example
- Characteristics do not select a post-crossing weak solution Remark
- Inviscid Burgers characteristic formula and first crossing time Theorem
- Uniqueness of a classical quasilinear solution before characteristic crossing Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- First order PDE: The Methods of Characteristics (standard reference, not scraped)