Foundations

Continuity preserves openness through inverse images.

Triangle-angle relations depend on the geometry.

Vector-space laws extend beyond coordinate arrows.

A basis spans without linear redundancy.

Kernel and image dimensions sum to domain dimension.

An eigenvector changes only by a scalar factor.

Self-adjoint operators have orthonormal eigenbases.

Any real rectangular matrix has orthogonal factors.

Real convergence can be checked by distances between late terms.

A series converges through its partial sums.

The remainder quantifies a finite polynomial approximation.

One cutoff controls the error at every domain point.

Integration starts from measurable simple functions.

Differentiability means a linear approximation with smaller-order error.

A continuous double integral can be evaluated by slicing.

These coordinate derivatives have different input and output types.

Circulation around a boundary equals oriented curl flux.

Outward boundary flux equals integrated divergence.

Local Lipschitz continuity ensures one local trajectory.

The matrix exponential solves a finite linear system.

A PDE becomes a solution problem only with a domain and data.

A distribution assigns probabilities to a random variable's values.

A normal model is specified by location and positive scale.

Different count mechanisms require different distributions.

A sample mean concentrates near its mean under stated assumptions.

Centered sample averages have an asymptotic normal law.

A conditional distribution renormalizes a joint law.

Likelihood is a function of a parameter given observed data.

A p-value is calibrated under a specified null model.

Selection and treatment assignment answer different questions.