Foundations

For y″+p(x)y′+q(x)y=f(x), use independent homogeneous solutions y₁,y₂ and set yₚ=u₁y₁+u₂y₂.

An n-element set has 2ⁿ subsets; its k-element subsets correspond to length-n binary strings with k ones.

For aₙ=saₙ₋₁+taₙ₋₂ with t≠0, distinct roots r₁,r₂ of r²−sr−t give aₙ=Ar₁ⁿ+Br₂ⁿ; a repeated root r gives aₙ=(A+Bn)rⁿ.

Residuals eᵢ=yᵢ−f(xᵢ;β̂) compare observed and fitted responses.

A chain satisfies P(Xₙ₊₁=j|X₀,…,Xₙ)=Pᵢⱼ when Xₙ=i, where P has nonnegative entries and each row sums to one.

A stationary distribution is a nonnegative row vector π with Σπᵢ=1 and πP=π.

A rate-λ Poisson counting process starts at zero, has independent increments and N(t+s)−N(s) distributed Poisson(λt).

Standard Brownian motion has B(0)=0, almost surely continuous paths and independent increments with B(t)−B(s)∼N(0,t−s) for 0≤s<t.

On a sorted random-access array, compare the target with the middle key and discard the half that cannot contain it; repeat until a matching key is found or the remaining interval is empty.

Split an array into two nearly equal halves, sort each recursively, and merge sorted halves by repeatedly choosing the next ordered key.

To maximize the number of compatible intervals, repeatedly take the available interval finishing earliest and discard overlapping intervals.

In a connected undirected weighted graph, if a set of chosen edges lies within some minimum spanning tree and a cut crosses none of them, a lightest edge crossing that cut can be added while preserving this invariant.

Process edges in nondecreasing weight order, adding an edge exactly when its endpoints belong to different current components, then merge those components.

For eventually nonnegative costs f,g, f=Θ(g) means c₁g(n)≤f(n)≤c₂g(n) for all sufficiently large n and constants c₁,c₂>0.

If operation i changes state Dᵢ₋₁ to Dᵢ, define ĉᵢ=cᵢ+Φ(Dᵢ)−Φ(Dᵢ₋₁).

Append to an array with spare capacity; when full, allocate twice the capacity and copy existing entries.

For a 3CNF formula with m clauses, make a vertex for each literal occurrence.

Over F₂ take H=[A|Iₘ] and G=[Iₙ₋ₘ;A].

For received r=c+e with Hc=0, syndrome Hr=He.

If columns of invertible P are an ordered basis B in standard coordinates, v=P[v]B.

From independent real vectors v₁,…,vₖ form wⱼ=vⱼ−Σ_(i<j)[(vⱼ·wᵢ)/(wᵢ·wᵢ)]wᵢ.

For uᵢ₊₁=uᵢ+hφ(tᵢ,uᵢ,h), if the exact-value one-step defect is bounded by Chᵖ⁺¹ and |∂ᵤφ|≤L, L>0, then exact initialization gives |u(tᵢ)−uᵢ|≤(Chᵖ/L)(eᴸ⁽ᵗⁱ⁻ᵃ⁾−1).

An invertible affine transformation of ℝⁿ has form T(x)=Ax+b with A invertible.

Every distance-preserving bijection of ℝⁿ has form T(x)=Qx+b with QᵀQ=I; conversely each such map is an isometry.

A bijection of ℝⁿ uniformly scaling all distances by s>0 is a similarity and has form T(x)=sQx+b with Q orthogonal.

The symmetries of a figure S⊂ℝⁿ are the Euclidean isometries g with g(S)=S.

For a one-to-one C¹ transformation T mapping a planar or spatial integration region G onto R, with nonzero interior Jacobian, ∫R f(x)dx=∫G f(T(u))|det DT(u)|du.

A regular C¹ plane or space curve r(t), a≤t≤b, has length ∫ₐᵇ‖r′(t)‖dt.

A C¹ regular parameterization r(u,v) has rᵤ×rᵥ≠0 and area element dS=‖rᵤ×rᵥ‖du dv.

The subgroup generated by a is ⟨a⟩={aᵏ:k∈Z}; its order is the smallest positive n with aⁿ=e, or infinite if none exists.