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.
Characteristics are covectors before they are curves
For higher-order equations, characteristic objects are first defined by the principal symbol acting on covectors, hence on conormals to hypersurfaces. For first-order transport the transport vector field directly produces familiar base-space characteristic curves. Higher-order real principal symbols can also produce curves: their Hamilton vector fields define bicharacteristics in the cotangent bundle, whose projections are characteristic rays. Thus the transport curve picture is a special direct construction, while the general higher-order curve picture retains the covector variable.
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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
- Gerald Teschl, Partial Differential Equations: From Classical to Modern (standard reference, not scraped)