f(x) = Σ a_ncos(nωx) + b_nsin(nωx)

F(ω) = ∫f(t)e^−iωtdt

f * g = ∫f(τ)g(t−τ)dτ

F[f * g] = F[f] · F[g]

a_n = (2/T)∫f(t)cos(nωt)dt

δ(t) = (1/2π)∫e^iωtdω

f(x) = Σ a_ncos(nωx) + b_nsin(nωx)

F(ω) = ∫f(t)e^−iωtdt

f * g = ∫f(τ)g(t−τ)dτ

F[f * g] = F[f] · F[g]

a_n = (2/T)∫f(t)cos(nωt)dt

δ(t) = (1/2π)∫e^iωtdω

p(θ|D) ∝ p(D|θ)p(θ)

Var(X) = 𝔼[X²] − (𝔼[X])²

P(A|B) = P(B|A)P(A)/P(B)

X ~ 𝒩(μ, σ²)

ℒ(θ) = Π p(x_i|θ)

MLE: ∂ log ℒ/∂θ = 0

p(θ|D) ∝ p(D|θ)p(θ)

Var(X) = 𝔼[X²] − (𝔼[X])²

P(A|B) = P(B|A)P(A)/P(B)

X ~ 𝒩(μ, σ²)

ℒ(θ) = Π p(x_i|θ)

MLE: ∂ log ℒ/∂θ = 0

H(X) = −Σ p(x) log p(x)

I(X;Y) = H(X) − H(X|Y)

DFT: X_k = Σ x_ne^−i2πkn/N

Parseval: Σ|x_n|² = Σ|X_k|²/N

KL(p ∥ q) = Σ p log(p/q)

∇ × E = −∂B/∂t

H(X) = −Σ p(x) log p(x)

I(X;Y) = H(X) − H(X|Y)

DFT: X_k = Σ x_ne^−i2πkn/N

Parseval: Σ|x_n|² = Σ|X_k|²/N

KL(p ∥ q) = Σ p log(p/q)

∇ × E = −∂B/∂t

Laplace: F(s) = ∫f(t)e^−stdt

z = F(s) poles → dynamics

W(ω) = |F(ω)|²

CLT: S_n → 𝒩(0,1)

χ² = Σ(O−E)²/E

e^iθ = cos θ + i sin θ

Laplace: F(s) = ∫f(t)e^−stdt

z = F(s) poles → dynamics

W(ω) = |F(ω)|²

CLT: S_n → 𝒩(0,1)

χ² = Σ(O−E)²/E

e^iθ = cos θ + i sin θ

Writing

A small set of notes and longer explanations.

Preprint

2026

Probabilistic Invertibility of Rectified Flows Beyond Global Monotonicity

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Least Squares and Likelihood in Linear Regression

A note on linear regression from the least-squares and likelihood points of view.

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