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.

Oriented simplicial cell chains and boundary

Definition

Let $T$ be a matching simplicial mesh as in Matching affine simplicial meshes and shape regularity. Set $C_{- 1} (T; Z) = 0$ . For $0 \leq k \leq d$ , let $C_{k} (T; Z)$ be the free abelian group generated by ordered $k$ -faces $[v_{0}, \dots, v_{k}]$ , with the orientation relation $[v_{π (0)}, \dots, v_{π (k)}] = sgn (π) [v_{0}, \dots, v_{k}]$ for every permutation $π$ . Define $\partial_{0} : C_{0} \to C_{- 1}$ to be the zero map. For $1 \leq k \leq d$ , define the simplicial boundary homomorphism $\partial_{k} : C_{k} \to C_{k - 1}$ on a generator by $\partial_{k} [v_{0}, \dots, v_{k}] : = \sum_{j = 0}^{k} (- 1)^{j} [v_{0}, \dots, {\hat{v}}_{j}, \dots, v_{k}] .$ This is the oriented incidence convention; no geometric integration is part of the definition. The identity $\partial_{k - 1} \partial_{k} = 0$ is proved directly from this alternating formula in the following lemma.

Depends on

Matching affine simplicial meshes and shape regularity

Used by

Dependency tree · one level

1 result within one dependency step 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

Jean-Luc Guermond, MATH 661 Fall 2025, Chapter 10, “Mesh orientation” (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 8 (standard reference, not scraped)