Foundations

On n equal subintervals with h=(b−a)/n, the trapezoidal estimate is T=h[f(a)/2+Σᵢ₌₁ⁿ⁻¹f(a+ih)+f(b)/2].

Euler stepping for y′=f(t,y) uses yₖ₊₁=yₖ+hf(tₖ,yₖ).

Set k₁=hf(t,y), k₂=hf(t+h/2,y+k₁/2), k₃=hf(t+h/2,y+k₂/2), k₄=hf(t+h,y+k₃); then y_next=y+(k₁+2k₂+2k₃+k₄)/6.

For y′=f(t)g(y) on a region where g(y)≠0, integrating dy/g(y)=f(t)dt gives an implicit solution.

For y′+p(t)y=q(t) with continuous p,q, μ(t)=exp(∫p(t)dt) transforms the equation into (μy)′=μq.

For ay″+by′+cy=0, a≠0, roots of ar²+br+c=0 give the solution modes: distinct real roots give eʳ¹ᵗ,eʳ²ᵗ; a repeated root gives eʳᵗ,teʳᵗ; conjugate α±iβ give eᵅᵗcos βt,eᵅᵗsin βt.

For a linear equation Ly=f, any one particular solution yₚ plus the general homogeneous solution y_c gives every solution y=y_c+yₚ.

In a C¹ planar autonomous system, equilibria are zeros of the vector field.

For a 2L-periodic piecewise smooth real f, aₙ=L⁻¹∫₋ᴸᴸf(t)cos(nπt/L)dt and bₙ=L⁻¹∫₋ᴸᴸf(t)sin(nπt/L)dt.

The one-sided Laplace transform is F(s)=∫₀∞e⁻ˢᵗf(t)dt.

For u_t=κu_xx on 0<x<L with zero endpoint temperatures, separation gives modes sin(nπx/L)e⁻κ⁽ⁿπ/L⁾²ᵗ.

For u_tt=c²u_xx with zero ends on [0,L], extend initial displacement f and velocity g oddly with period 2L to F,G.

A hash table maps keys to buckets and stores colliding keys in a chain.

An AVL binary search tree maintains subtree heights differing by at most one at every node.

A binary max heap stores a complete binary tree in an array with every parent key at least its children's keys.

Build a max heap, exchange its root with the last active element, shrink the active heap and restore heap order; repeat.

Dynamic programming states subproblems and recurrences, checks an acyclic dependency order and base cases, and evaluates each needed subproblem once by memoization or bottom-up order.

For positive integers a₁,…,aₙ and target T≥0, D(i,t)=D(i−1,t) or [t≥aᵢ and D(i−1,t−aᵢ)], with D(0,0)=true and other initial states false.

Initialize d(s)=0 and other distances to infinity, then relax every directed edge for |V|−1 rounds.

If C¹ f:U⊆Rⁿ→Rⁿ has invertible derivative at a∈U, it restricts to a C¹ bijection between neighborhoods of a and f(a), with C¹ inverse and D(f⁻¹)(f(a))=[Df(a)]⁻¹.

For C¹ F:U⊆Rⁿ×Rᵐ→Rᵐ with F(a,b)=0 and invertible partial derivative D_yF(a,b), nearby zeros are uniquely y=g(x), where g is C¹ and g(a)=b.

A sigma-algebra contains the whole space and is closed under complements and countable unions.

A map between measurable spaces is measurable when every measurable target set has a measurable inverse image.

On sigma-finite measure spaces, a nonnegative product-measurable function has equal product and iterated integrals, possibly infinite.

A topological space is connected when it cannot be partitioned into two nonempty disjoint open sets.

A metric space is complete when every Cauchy sequence has a limit in that space.

A self-map T of a nonempty complete metric space with d(Tx,Ty)≤k d(x,y), 0≤k<1, has one fixed point.

For square-integrable real X,Y, Var X=E[(X−EX)²] and Cov(X,Y)=E[(X−EX)(Y−EY)].

A joint mass or density specifies simultaneous values; marginals sum or integrate out the other coordinates.

E[X|Y=y] averages X using its conditional law.