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.

Applied Materials - Q2 FY2026 Earnings $AMAT 461.85 [+4.8% AH] ✅ Revenue: $7.91B (Est: $7.68B) [+11% YoY] ✅ Adj. EPS: $2.86 (Est: $2.68) Additional Metrics: Non-GAAP Gross Margin: 50.0% Non-GAAP Operating Margin: 32.1% Semiconductor Systems Revenue: $5.97B Semiconductor Systems Non-GAAP Operating Margin: 35.2% Applied Global Services Revenue: $1.67B Applied Global Services Non-GAAP Operating Margin: 29.2% Q3 FY2026 Guidance: ⚪ Revenue: $8.45-9.45B (Est: $8.15B) ⚪ Adj. EPS: $3.16-3.56 (Est: $2.90)

APPLIED MATERIALS $AMAT JUST REPORTED Q2 EARNINGS Revenue: $7.91B vs $7.65B est 🟢 Adj EPS: $2.86 vs $2.66 est 🟢 Adj Net Income: $2.29B vs $2.11B est 🟢 Adj Gross Margin: 50% Adj Operating Margin: 32.1% Adj Free Cash Flow: $210M GAAP figures: EPS: $3.51 Net Income: $2.81B

Apple’s two-year-old partnership with OpenAI has become strained, according to people familiar with the matter, with the AI startup failing to see the expected benefits from the deal and now preparing possible legal action. @markgurman reports

Apple’s $AAPL two-year partnership with OpenAI has become strained OpenAI has reportedly failed to see the expected benefits from the deal and now preparing possible legal action - Bloomberg