Manuscript XXII-D@1 · XXII-D.tex · assumption · 17889d18

Model specification: Conditional extraction output

Conditional extraction and calibration · lines 326–357

XXII-D@1#spec:extraction, XXII-D@1#eq:faithful, XXII-D@1#eq:span, XXII-D@1#eq:generated, XXII-D@1#eq:lorentz

Fix $m \geq 3$ . For a datum $D$ , suppose a predeclared rule has supplied

(V_{D}, ι_{D}, K_{D}, T_{D})

with $dim V_{D} = m$ , $ι_{D} : E_{D} \to V_{D}$ injective, $K_{D}$ proper, and $T_{D} : V_{D} \to R^{m}$ invertible, such that

p ⪯_{D} q V_{D} K_{D} T_{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}, = L_{m} = {(t, x) \in R \times R^{m - 1} : t \geq ∥ x ∥_{2}} .

The construction must satisfy the non-circularity and admissibility audits of XXII-A and XXII-C. Neither its existence nor a nonempty admissible domain is asserted in this specification.

Displayed formulas in this passage

XXII-D@1#eq:faithful · line 335
$p ⪯_{D} q V_{D} K_{D} T_{D} (K_{D}) ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q}, = L_{m} = {(t, x) \in R \times R^{m - 1} : t \geq ∥ x ∥_{2}} .$
XXII-D@1#eq:span · line 335
$p ⪯_{D} q V_{D} K_{D} T_{D} (K_{D}) ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q}, = L_{m} = {(t, x) \in R \times R^{m - 1} : t \geq ∥ x ∥_{2}} .$
XXII-D@1#eq:generated · line 335
$p ⪯_{D} q V_{D} K_{D} T_{D} (K_{D}) ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q}, = L_{m} = {(t, x) \in R \times R^{m - 1} : t \geq ∥ x ∥_{2}} .$
XXII-D@1#eq:lorentz · line 335
$p ⪯_{D} q V_{D} K_{D} T_{D} (K_{D}) ⟺ ι_{D} (q) - ι_{D} (p) \in K_{D}, = span {ι_{D} (q) - ι_{D} (p) : p, q \in E_{D}}, = cone {ι_{D} (q) - ι_{D} (p) : p ⪯_{D} q, p \neq = q}, = L_{m} = {(t, x) \in R \times R^{m - 1} : t \geq ∥ x ∥_{2}} .$
unlabeled · line 330
$(V_{D}, ι_{D}, K_{D}, T_{D})$

Nearby in this section

section: Conditional extraction and calibration