Manuscript XXII-A@1 · XXII-A.tex · definition · 9e4e9f66

Definition: Faithful generated affine representation

The frozen target and its acceptance conditions · lines 494–521

XXII-A@1#def:representation, XXII-A@1#eq:faithful, XXII-A@1#eq:span, XXII-A@1#eq:generated

Let $E_{D}$ be the declared event carrier associated with operational datum $D$ , and let $⪯_{D}$ be a declared partial order extracted from scheduled ancestry or operational influence.

A faithful generated affine cone representation consists of $(V_{D}, ι_{D}, K_{D})$ , where $V_{D}$ is a finite-dimensional real vector space, $ι_{D} : E_{D} \to V_{D}$ is injective, and $K_{D}$ is a proper cone satisfying

p ⪯_{D} q V_{D} K_{D} ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q} .

If the carrier is obtained by an operational quotient or completion, that construction and the relation on the resulting carrier are part of the extraction.

Displayed formulas in this passage

XXII-A@1#eq:faithful · line 504
$p ⪯_{D} q V_{D} K_{D} ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q} .$
XXII-A@1#eq:span · line 504
$p ⪯_{D} q V_{D} K_{D} ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q} .$
XXII-A@1#eq:generated · line 504
$p ⪯_{D} q V_{D} K_{D} ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q} .$

Nearby in this section

section: The frozen target and its acceptance conditions