Computer Science core problems

Problem 1 – Resonant algorithm runtime#

A triadic algorithm runs in

$$ T = D_6 τ_r \log(X), $$

where $$D_6$$ is a structural triad, $$τ_r$$ is resonant-time, and $$X = F_3 T_f$$.

If $$τ_r$$ is halved, how does the runtime $$T$$ change?

Problem 2 – Data throughput in a resonant bus#

A resonant-bus architecture has throughput

$$ R = F_3 T_f D_3. $$

If $$D_3$$ increases by 2 (i.e., triples) and $$T_f$$ decreases by 10%, what is the net effect on $$R$$?

Problem 3 – Error correction under resonance#

Error probability in a resonant communication channel is

$$ p = e^{-ΛΘ τ_r}. $$

If the system requires $$p$$ to be reduced by 50%, what change to $$τ_r$$ is needed, assuming $$ΛΘ$$ is fixed?