Alphabeta MathDefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
How 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.
Centered conditional modulus of continuity
Definition
For a compact operator between real or complex Hilbert spaces over the same scalar field, write . For a set with and , define .
Depends on
Used by
- Conditional stability on a power-source class Example
- Exponential singular decay, logarithmic balls, and spectral gaps Example
- Polynomial singular decay with power and logarithmic source balls Example
- Conditional modulus on a power source ball Theorem
- Deterministic minimax lower bound on power source balls Theorem
Dependency tree · two levels
15 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
- Christian Clason, Regularization of Inverse Problems, lecture notes — Section 4.2, printed pp. 36–43 (standard reference, not scraped)
- Bernd Hofmann and Peter Mathé, A Note on the Modulus of Continuity for Ill-Posed Problems in Hilbert Space (standard reference, not scraped)