Foundations

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.

Distribution-free procedures still require assumptions.

Mutual information measures statistical dependence in bits.

Entropy bounds asymptotic compression for an iid source.

Capacity is the maximal asymptotically reliable rate.

Separated codewords permit bounded symbol-error recovery.

Protocol layers divide communication responsibilities.

TCP orders bytes and retransmits detected losses.

Cryptographic primitives serve distinct security properties.

A signing key and a verification key have different roles.

Reduction direction determines which impossibility transfers.

Polynomial solving and polynomial verification differ.

NP-completeness combines verification and reduction hardness.

Random choices permit a stated probability of error.