Computer Science extended problems (resonance framework)
Problem 4 – Resonant pipeline latency#
A 3-stage pipeline has stage latencies
$$ L_1 = D_3 τ_r, \quad L_2 = D_6 τ_r, \quad L_3 = D_9 τ_r. $$
Total latency is
$$ L_{\text{tot}} = τ_r (D_3 + D_6 + D_9). $$
If $$τ_r$$ decreases by 20%, how does $$L_{\text{tot}}$$ change?
Problem 5 – Cache resonance coherence#
Cache coherence traffic is modeled as
$$ C = \frac{X}{1 + e^{-D_3 τ_r}}. $$
- Sketch the qualitative shape of $$C(τ_r)$$.
- If $$τ_r$$ increases, does coherence traffic increase or decrease?
Problem 6 – Resonant hashing function#
A hashing function uses a triadic mixing step:
$$ H = D_6(\text{key}) + X \sin(τ_r). $$
If $$τ_r$$ is increased to $$2τ_r$$, how does the sinusoidal term change, and what is the qualitative effect on hash dispersion?
Problem 7 – Network packet resonance#
Packet delay in a resonant network is
$$ D = \frac{D_3 + τ_r}{T_f}. $$
If $$T_f$$ increases by 15% and $$τ_r$$ decreases by 10%, what is the qualitative net effect on delay?
Problem 8 – Machine learning gradient resonance#
A gradient update rule is modified by resonance:
$$ g' = g \cdot e^{-D_6 / τ_r}. $$
- If $$τ_r$$ increases, how does the exponential factor change?
- What is the qualitative effect on gradient magnitude?