Research proof index

Characterization of the order on Bohr sets. The relation $B_1\le B_2$ holds exactly when every frequency of $B_2$ is also a frequency of $B_1$, and $\operatorname{width}_{B_1}(\psi)\le \operatorname{width}_{B_2}(\psi)$ for each such frequency.

A point $x$ belongs to the chord model of a Bohr set $B$ exactly when $\|1-\psi(x)\|\le \operatorname{width}_B(\psi)$ for every frequency $\psi$ of $B$.

NOTE: This could be generalized to 2 * N instead of 2 ^ (ϑ + 1). Also, this can be proved for ↔ instead of →.

General entropy non-negativity under a pointwise upper bound probability ≤ 1. This assumption holds automatically in the finite/counting case (since sums bound each term), but can fail in general (continuous) settings; hence we separate it as a hypothesis. Finite case: see CanonicalEnsemble.entropy_nonneg in Finite.

Solving the equation of motion for the acceleration: along a solution the second derivative is -(k/m) x - (γ/m) ẋ.

The variational gradient of the Caldirola–Kanai action is the exponential factor times the difference of the force and mass times acceleration appearing in Newton's second law.