Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · section · 8ca47088

A bounded segment is new comparison data

Bounded comparisons and the separate ramp prescription › A bounded segment is new comparison data · lines 1898–1924

If a separate restriction $∣ λ ∣ \leq Λ$ is imposed, let $d = Λ c_{ε} > 0$ . Direct amplitude readout obeys

∣ λ ∣ \leq Λ sup d_{TV} (Law_{read} (λ c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (d / σ) .

For $0 < δ, v < 1/2$ and a separately specified benchmark magnitude $A_{sig} > 0$ , variation at most $δ$ and visibility at least $v$ hold exactly when

\frac{d}{artanh ( 2 δ )} \leq σ \leq \frac{A _{sig}}{artanh ( 2 v )} .

These statements follow by monotonicity of $tanh$ XEnter.

Compact support gives no such amplitude restriction: every real multiple of a supported vector has the same support. A bounded comparison set also need not be an algebra closed under composition.

Displayed formulas in this passage

unlabeled · line 1903
$∣ λ ∣ \leq Λ sup d_{TV} (Law_{read} (λ c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (d / σ) .$
unlabeled · line 1912
$\frac{d}{artanh ( 2 δ )} \leq σ \leq \frac{A _{sig}}{artanh ( 2 v )} .$