DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Convex invariant domains for a one-step update

Definition

Let $m, N \geq 1$ , let $A \subseteq R^{m}$ be a state domain, and let $Φ : A^{N} \to A^{N}$ be a one-step update on $N$ cell states. A set $K \subseteq A$ is convex if $θ U + (1 - θ) V \in K$ for every $U, V \in K$ and $θ \in [0, 1]$ . It is a convex invariant domain for $Φ$ if it is convex and $Φ (K^{N}) \subseteq K^{N}$ . The word invariant refers only to this stated one-step map and its stated admissible inputs; it asserts no invariance for a different flux, time integrator, or exact Riemann evolution.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

J.-L. Guermond and B. Popov, Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems (standard reference, not scraped)