Alphabeta Math
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.

Periodic scalar cell problem

Definition

For coefficient a, cell Y, and zero-mean periodic space V# from Uniformly elliptic periodic scalar diffusion data and Zero-mean periodic Sobolev cell space, and for p∈Rd, the weak periodic cell equation asks for χp∈V# such that ∫Ya(y)(p+∇yχp(y))⋅∇yv(y) dy=0(v∈V#). The sign convention is p+∇yχp; the zero-mean condition fixes the additive constant. This definition alone asserts neither existence nor uniqueness.

Remarks

Existence and uniqueness of periodic cell correctors supplies existence, uniqueness, linearity, and energy bounds for this definition.

Depends on

Used by

Dependency tree · two levels

6 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