DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Complement splitting and range projection for a simple fold

Definition

Let $J : R^{n} \to R^{n}$ be linear of rank $n - 1$ (Rank and nullity of a linear map with finite-dimensional domain). Let $v$ be a unit vector with $\ker J = span {v}$ , let $w \neq 0$ satisfy $w^{T} J = 0$ , and let $b \in R^{n}$ satisfy $a : = w^{T} b \neq 0$ . Define the state-space complement and range complement by $X : = v^{⊥}, Y : = \ker (w^{T}) .$ Set $q : = b / a$ and define the projection candidate $P : = I - q w^{T} : R^{n} \to R^{n} .$ For a differentiable $F : U \subseteq R^{n} \times R \to R^{n}$ , $F_{x}$ and $F_{λ}$ denote the restrictions of the total derivative The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder to the state and scalar-parameter directions. At a simple fold, the intended data are $J = F_{x} (x_{0}, λ_{0})$ and $b = F_{λ} (x_{0}, λ_{0})$ . The next lemma proves that $J ∣_{X} : X \to Y$ is an isomorphism and that $P$ is the projection onto $Y$ along $span {q}$ .

Depends on

Used by

Dependency tree · two levels

17 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

Eugene L. Allgower and Kurt Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics 13 (1990) (standard reference, not scraped)
K. A. Cliffe, A. Spence, and S. J. Tavener, The numerical analysis of bifurcation problems with application to fluid mechanics, Acta Numerica 9 (2000), 39–131 (standard reference, not scraped)