ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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 Dirichlet Laplacian inverse is uniformly bounded in the maximum norm

Example

For $h = 1 / N$ , $N \geq 2$ , let $A_{N}$ act on vectors $(v_{1}, \dots, v_{N - 1})$ with $v_{0} = v_{N} = 0$ by

(A_{N} v)_{i} = \frac{2 v_{i} - v_{i - 1} - v_{i + 1}}{h^{2}}, 1 \leq i < N .

With the vector maximum norm $∥ v ∥_{\infty} = \max_{1 \leq i < N} ∣ v_{i} ∣$ , the inverse satisfies $∥ A_{N}^{- 1} ∥_{\infty \to \infty} \leq 1 / 8$ for every $N \geq 2$ .

Facts & Assumptions

Given: $N \geq 2$ , $h = 1 / N$ , $v_{0} = v_{N} = 0$ , and the displayed matrix action of $A_{N}$ .

[A1]

The grid vector norm is $∥ v ∥_{\infty} = \max_{1 \leq i < N} ∣ v_{i} ∣$ , and the matrix uses zero endpoint values.

Verification

1.1givenA1algebra

Set $x_{i} = i h$ and $ϕ_{i} = x_{i} (1 - x_{i}) / 2$ for $0 \leq i \leq N$ . Then $ϕ_{0} = ϕ_{N} = 0$ , $- δ_{h}^{2} ϕ_{i} = 1$ , so $A_{N} ϕ = 1$ , and $0 \leq ϕ_{i} \leq 1 / 8$ because $x (1 - x) \leq 1 / 4$ on $[0, 1]$ .

1.2givenA1algebra

If $w_{0} = w_{N} = 0$ and $A_{N} w \geq 0$ at every interior index, a negative interior minimum $w_{k} < 0$ would give $(A_{N} w)_{k} = (2 w_{k} - w_{k - 1} - w_{k + 1}) / h^{2} \leq 0$ . Equality is then forced, so both neighboring values equal $w_{k}$ ; repeating this along the finite grid reaches a zero endpoint, a contradiction. Therefore $w_{i} \geq 0$ everywhere.

2.1givenA1step 1.2algebra

The kernel of $A_{N}$ is zero: if $A_{N} v = 0$ and $v$ has a positive interior maximum, the equation forces both neighboring values to equal that maximum, which propagates to a zero endpoint; applying the same argument to $- v$ rules out a negative minimum. Thus $v = 0$ , and the square finite matrix is invertible.

3.1step 1.1step 1.2step 2.1A1algebra∎

Let $g$ be any right-hand side, let $v = A_{N}^{- 1} g$ , and put $M = ∥ g ∥_{\infty}$ . If $M = 0$ , then $v = 0$ . If $M > 0$ , $A_{N} (M ϕ + v) = M 1 + g \geq 0$ and $A_{N} (M ϕ - v) = M 1 - g \geq 0$ ; step 1.2 makes both grid functions nonnegative. Hence $∣ v_{i} ∣ \leq M ϕ_{i} \leq M / 8$ for each $i$ , proving $∥ A_{N}^{- 1} g ∥_{\infty} \leq ∥ g ∥_{\infty} / 8$ .

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.