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.

Total-degree and hyperbolic-cross chaos index sets

Definition

For $d$ inputs with admissible cutoff vector $p = (p_{1}, \dots, p_{d})$ from Finite product-law polynomial-chaos input data, the cutoff total-degree set of order $s \geq 0$ and cutoff hyperbolic-cross set of threshold $N \geq 1$ are $Λ_{s, p}^{T D} = {α \in N_{0}^{d} : 0 \leq α_{j} \leq p_{j}, \sum_{j = 1}^{d} α_{j} \leq s}, Λ_{N, p}^{H C} = {α \in N_{0}^{d} : 0 \leq α_{j} \leq p_{j}, \prod_{j = 1}^{d} (α_{j} + 1) \leq N},$ The cutoff vector prevents either set from requesting polynomials above the moment degree declared in Finite product-law polynomial-chaos input data. Both are finite and contain zero. An arbitrary finite index set used below is denoted by $Λ$ ; whenever a downward-closed set is required, this means $α \in Λ$ and $0 \leq β_{j} \leq α_{j}$ for every $j$ imply $β \in Λ$ .

Depends on

Finite product-law polynomial-chaos input data

Used by

Dependency tree · two levels

3 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

Bruno Sudret, Polynomial Chaos Expansions and Stochastic Finite Element Methods, §3.3.2 (standard reference, not scraped)
Dongbin Xiu, Fast Numerical Methods for Stochastic Computations, §3 (standard reference, not scraped)