ExampleConstruction: Literature-sourcedVerification: Not suppliedjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

A locally finite hat-function partition of unity on $R$ subordinate to overlapping intervals

Example

For $n \in Z$ , let $ψ (t) = \max {1 - ∣ t ∣, 0}$ and $φ_{n} (x) = ψ (x - n)$ . The functions are continuous, take values in $[0, 1]$ , and have support $[n - 1, n + 1]$ . They are subordinate to the open cover $U_{n} = (n - \frac{3}{2}, n + \frac{3}{2})$ of $R$ .

If $x \in [n, n + 1]$ , the only possibly nonzero functions are $φ_{n}$ and $φ_{n + 1}$ , and $φ_{n} (x) + φ_{n + 1} (x) = (1 - (x - n)) + (1 - (n + 1 - x)) = 1.$ Thus ${φ_{n}}_{n \in Z}$ is a locally finite partition of unity subordinate to ${U_{n}}$ . The support intervals show local finiteness directly: every bounded set meets only finitely many support intervals.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

J. Robbin, Partitions of Unity (standard reference, not scraped)
K. Datchev, Iterated interpolation and a partition of unity (Purdue University) (standard reference, not scraped)

A locally finite hat-function partition of unity on RR subordinate to overlapping intervals