Phase tail sets for amplification

PDF p. 22: Section 4, equation (4.1)

Let $V$ be a nonempty finite set of vertices, and let $\theta \in T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$ be a phase configuration: each coordinate is a real angle modulo $2\pi$. For each $x\in V$, write ${\theta }_x$ for its unique representative in $(-\pi ,\pi ]$. For an angle $0\le \psi \le \pi$, define the phase tail set

${\mathcal{C}}_{\psi }(\theta ):=\{x\in V:\cos {\theta }_x\le \cos \psi \}=\{x\in V:\mathrm{\mid }{\theta }_x\mathrm{\mid }\ge \psi \}\mathrm{.}$

Thus ${\mathcal{C}}_{\psi }(\theta )$ consists of the vertices whose principal angular distance from phase zero is at least $\psi$; here that distance is $\mathrm{\mid }{\theta }_x\mathrm{\mid }$.

Finite phase tori, local lifts and the flat metric

Phasors, trigonometric identities and periodic derivatives

Angular order and elementary inverse-trigonometric bounds

Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1

Given: A phase configuration $\theta$ indexed by the nonempty finite set $V$ and an angle $\psi \in [0,\pi ]$.

Goal: Verify the representative convention and the equality of the two descriptions of ${\mathcal{C}}_{\psi }(\theta )$.

1.1 For a real lift of a coordinate, choose the representative $r\in [0,2\pi )$ from [F2]. Use $r$ if $r\le \pi$ and $r-2\pi$ otherwise; this lies in $(-\pi ,\pi ]$ and represents the same phase by [F1]. Two representatives in that interval have difference of absolute value less than $2\pi$, so [F1] makes them equal if they represent the same phase. By [F5], $\mathrm{\mid }{\theta }_x\mathrm{\mid }\in [0,\pi ]$. Any other representative is ${\theta }_x+2\pi k$. For $k\ge 1$ it is greater than $\pi$, and for $k\le -1$ it is at most $-\pi$; its absolute value is therefore at least $\mathrm{\mid }{\theta }_x\mathrm{\mid }$. Thus $\mathrm{\mid }{\theta }_x\mathrm{\mid }$ is the least absolute value among real representatives of that phase. The cosine condition is also independent of the chosen lift by periodicity. [F1, F2, F3, F5, algebra]

2.1 Taking real parts in [F3] gives $\cos (-u)=\cos u$, so $\cos {\theta }_x=\cos \mathrm{\mid }{\theta }_x\mathrm{\mid }$. Since both $\mathrm{\mid }{\theta }_x\mathrm{\mid }$ and $\psi$ lie in $[0,\pi ]$, strict decrease in [F4] gives $\cos \mathrm{\mid }{\theta }_x\mathrm{\mid }\le \cos \psi$ exactly when $\mathrm{\mid }{\theta }_x\mathrm{\mid }\ge \psi$. This holds for every vertex, proving the equality of the sets, including the endpoint cases. [F3, F4, F5, step 1.1] ∎

The definition and its equivalent description are attributed to Expander graphs are globally synchronizing, PDF page 22, Section 4, equation (4.1) and the following sentence; the principal representative convention is stated on PDF page 11. The justification above uses the mathematical suppliers, not the article as proof.

The definition and its equivalent description are attributed to Expander graphs are globally synchronizing, PDF page 22, Section 4, equation (4.1) and the following sentence; the principal representative convention is stated on PDF page 11. The justification above uses the mathematical suppliers, not the article as proof.