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.

Conservative nonnegative network advection routing

Definition

At a directed network junction, let $n_{i n}$ edges carry nonnegative incoming mass fluxes $q_{i}^{i n}$ into the junction, and let $n_{o u t}$ edges carry outgoing mass fluxes. A conservative nonnegative routing matrix is a real matrix $P \in R^{n_{o u t} \times n_{i n}}$ such that

$P_{j i} \geq 0, \sum_{j = 1}^{n_{o u t}} P_{j i} = 1 (1 \leq i \leq n_{i n}) .$

It routes the incoming flux vector to $q^{o u t} = P q^{i n}$ . The column $i$ distributes the mass flux arriving on incoming edge $i$ among the outgoing edges. The column-sum condition stipulates no source or sink at the junction. It is a model condition on flux; it does not assert that the densities on incident edges are equal.

Depends on

Used by

Density continuity can violate conservative advection balance Counterexample
A conservative routing matrix preserves total flux Theorem

Dependency tree · two levels

8 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

Ward Whitt, Continuous-Time Markov Chains, full lecture notes (2006) (standard reference, not scraped)