Approximating the Heaviside step function & its derivative with sigmoids Top: Smooth approximations to H(x) = 0 (x < 0), 1 (x > 0) using sigmoid(x; σ) = 1 / (1 + e^{-σ x}) for σ = 1 (orange), 2 (green), 4 (red), 8 (purple). Larger σ → sharper step. Bottom: Their derivatives d/dx [sigmoid] = σ ⋅ sigmoid(x; σ) ⋅ (1 − sigmoid(x; σ)) which converge to the Dirac delta δ(x) as σ → ∞. This Enables gradient-based optimization in neural networks and physics simulations where the true step function is non-differentiable.

H.E. Yasir Al-Rumayyan, Governor of PIF, participated in a fireside chat with Michael O’Grady, Chairman and CEO of Northern Trust, at the Institutional Investor Conference. The discussion highlighted the importance of long-term capital and how institutional investors are navigating the evolving global landscape. It builds on PIF’s partnerships with leading asset management firms, which are unlocking new investment opportunities and strengthening capabilities.