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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.
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H.B. FULLER’S BOARD WILL CAREFULLY EVALUATE THE PROPOSAL WITH ITS FINANCIAL AND LEGAL ADVISORS.
H.B. FULLER CONFIRMS RECEIPT OF AN UNSOLICITED $1.10–$1.20 BILLION CASH PROPOSAL FROM ANCORA TO ACQUIRE ITS BUILDING ADHESIVE SOLUTIONS BUSINESS.
@h_joshua_rivera of @blockchain_cap (8 years in, ~200 investments, ~$5B AUM) expanded the frame from agentic trading to agentic commerce, the whole agent economy learning to move money
H A Y L E Y Q U I N N @hayleyquinnxx loves BBC @TonyRopeBBC Tony Rope PAWGED
H-Town's QB 🤘
@CJ7STROUD | @HoustonTexans
h-town repping sactown now 👑
.@h_evva_n's Listening Party's gonna be unforgettable 😌
RSVP at the link below to join the Listening Party hosted by EVAN!
June 26, 15:00 KST:
*available only for Premium users in select regions
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