LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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.

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A second moment controls the discarded mass and number tails

Statement

For a positive measure $μ$ with finite second moment and every $R > 0$ , $\int_{(R, \infty)} x μ (d x) \leq \frac{M_{2} (μ)}{R}, μ ((R, \infty)) \leq \frac{M_{1} (μ)}{R} .$ In particular, for the bounded-kernel solution of Bounded-kernel measure solutions conserve mass and control number and second moments, these tails are uniform on every finite time interval after substituting the theorem's mass and second-moment bounds.

Facts & Assumptions

Given: A positive measure and the moment definitions on $E = [a, \infty)$ .

[F1]

Continuous-size coagulation-fragmentation weak model: the state measure is positive.

[F2]

Bounded-kernel measure solutions conserve mass and control number and second moments: on each finite interval, $M_{1}$ is constant and $M_{2} (t) \leq M_{2, 0} + K_{*} M_{1}^{2} t$ .

Proof

1.1givenF1

On $(R, \infty)$ , $x \leq x^{2} / R$ . Integrating this pointwise inequality against the positive measure gives $\int_{(R, \infty)} x d μ \leq M_{2} (μ) / R$ .

2.1givenF1F2algebra∎

On the same set, $1 \leq x / R$ . Integration gives $μ ((R, \infty)) \leq M_{1} (μ) / R$ . Applying [F2] makes both right-hand sides uniform for $t$ in a fixed finite interval.

Depends on

Used by

Finite cutoff differs from the unbounded pivot model by a controlled return error Lemma

Dependency tree · two levels

5 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

Philippe Laurençot, Weak compactness techniques and coagulation equations, §§1 and 3 (standard reference, not scraped)
Philippe Laurençot and Stéphane Mischler, From the discrete to the continuous coagulation-fragmentation equations, §§1–2 (standard reference, not scraped)