Kernel for stability amplification

PDF p. 16: Section 2.2, equation (2.13)

For real angles $\alpha ,\beta \in (-\pi ,\pi ]$, define the kernel, meaning the real-valued function $K:(-\pi ,\pi ]\times (-\pi ,\pi ]\to \mathbb{R}$, by $K(\alpha ,\beta )=\sin (\mathrm{\mid }\alpha \mathrm{\mid }-\min \{\mathrm{\mid }\beta \mathrm{\mid },\pi \mathrm{/}2\})$. Here $\mathrm{\mid }t\mathrm{\mid }$ is the absolute value of a real number and $\min \{a,b\}$ is the smaller of two real numbers. Sine and cosine are the real trigonometric functions.

Equivalently, $K(\alpha ,\beta )=\sin (\mathrm{\mid }\alpha \mathrm{\mid }-\mathrm{\mid }\beta \mathrm{\mid })$ when $\mathrm{\mid }\beta \mathrm{\mid }\le \pi \mathrm{/}2$, and $K(\alpha ,\beta )=-\cos \alpha$ when $\mathrm{\mid }\beta \mathrm{\mid }\ge \pi \mathrm{/}2$.

For every real $t$, absolute value is $t$ when $t\ge 0$ and $-t$ otherwise (Absolute value in an ordered field). Thus $\mathrm{\mid }\alpha \mathrm{\mid }$ and $\mathrm{\mid }\beta \mathrm{\mid }$ are real, and $\mathrm{\mid }\alpha \mathrm{\mid }$ is either $\alpha$ or $-\alpha$. The real sine function is defined for every real argument, so the displayed rule assigns a unique real number to every pair in the stated domain (Phasors, trigonometric identities and periodic derivatives). If $\mathrm{\mid }\beta \mathrm{\mid }\le \pi \mathrm{/}2$, the minimum is $\mathrm{\mid }\beta \mathrm{\mid }$, giving the first branch. If $\mathrm{\mid }\beta \mathrm{\mid }\ge \pi \mathrm{/}2$, the minimum is $\pi \mathrm{/}2$. The quarter-turn identity and oddness of sine give $\sin (\mathrm{\mid }\alpha \mathrm{\mid }-\pi \mathrm{/}2)=-\sin (\pi \mathrm{/}2-\mathrm{\mid }\alpha \mathrm{\mid })=-\cos (\mathrm{\mid }\alpha \mathrm{\mid })$; evenness of cosine then gives $-\cos (\mathrm{\mid }\alpha \mathrm{\mid })=-\cos \alpha$ (Quarter-turn values and shifts by pi/2 and pi, Parity and the Pythagorean identity for sine and cosine). At equality $\mathrm{\mid }\beta \mathrm{\mid }=\pi \mathrm{/}2$, both branches therefore have the same value.

Equation (2.13) of Expander graphs are globally synchronizing, PDF page 16, attributes this definition.