Mathematics

Vectors encode states and directions, while matrices encode linear input–output rules. This chapter derives matrix multiplication from composition, explains rows, columns, units, order, and linear systems, and separates the mathematical operation from algorithms that compute it.

Elementary algebra becomes reliable when a function is treated as a domain-aware mapping, graph transformations are read from the input outward, and every equation step is classified as reversible or implication-only. This chapter connects those habits across polynomial, rational, radical, exponential, logarithmic, trigonometric, and piecewise functions.

Inequality proofs are controlled transformations of order, not a bag of tricks. This chapter organizes sign-safe algebra, norm and mean inequalities, convexity, equality cases, and the process of choosing a bound that preserves the information the problem needs.

Limits turn “gets close” into a quantified contract; continuity lets local limits control whole intervals, while asymptotic notation compares rates without pretending two functions are equal. This chapter develops the definitions, calculation tools, error-aware examples, and conditions behind the standard shortcuts.

Derivatives measure local sensitivity and integrals measure signed accumulation; the Fundamental Theorem of Calculus explains why these apparently different limits invert one another. This chapter derives the core rules, states their domain conditions, and builds a practical workflow for differentiation, qualitative analysis, and integration.

A Taylor polynomial is the unique polynomial that matches a function’s derivatives at a chosen center, and its remainder turns local approximation into a quantitative contract. This chapter derives the coefficients, distinguishes finite approximations from infinite series and asymptotic expansions, and shows how to choose, bound, and evaluate an approximation safely.

A sequence converges when its tail enters every requested tolerance band, and a series converges only when its sequence of finite partial sums does. This chapter develops proof tools, convergence tests, error bounds, recurrence reasoning, and power-series domains without treating infinite notation as ordinary finite algebra.

Differentiability as local linearization, with gradients, chain rules, tangent geometry, Taylor models, and constrained optimization under explicit regularity conditions.

Double and triple integrals as accumulation over regions, with iterated bounds, Jacobians, polar/cylindrical/spherical coordinates, and checks that prevent region and orientation errors.

Line and surface integrals, conservative fields, divergence and curl, and the orientation-sensitive boundary theorems that connect local derivatives to global circulation and flux.

Initial-value problems, first-order methods, linear higher-order equations, forcing and resonance, and linear systems, with explicit checks for lost solutions, singular models, and uniqueness.

Complex differentiability requires one derivative limit to agree along every direction, turning the Cauchy–Riemann equations into a rigidity condition. This chapter connects the derivative to local rotation and scaling, harmonic conjugates, and the branch choices required for logarithms and complex powers.

Contour integration combines parameterized complex line integrals with holomorphic rigidity. Cauchy's theorem and integral formula turn boundary data into interior values, Laurent series classify isolated singularities, and residues reduce closed-contour integrals to local coefficients.

How IEEE 754 encodes reals in a fixed number of bits — exponent bias, normals and subnormals, signed zero, infinities and NaN — and what falls out of it: machine epsilon, five rounding modes with ties-to-even as default, non-associative addition, the exact bit-level reason 0.1 + 0.2 misses 0.3, FMA's single rounding, and the ~30-cycle cost of a subnormal on real silicon.

A Fourier series is an orthogonal projection of a periodic function onto sinusoidal modes. This chapter derives the coefficients, separates mean-square from pointwise convergence, explains symmetry, Parseval energy, Gibbs behavior, and coefficient decay, and keeps normalization and phase conventions explicit.

The Laplace transform adds exponential damping so growth, derivatives, delays, and convolution become algebraic in a complex variable. This chapter keeps the region of convergence explicit, derives the derivative and shift rules, explains inversion, and works a complete initial-value problem without misusing the delay theorem.

Fourier analysis rewrites a signal as amplitudes and phases of complex sinusoids. This entry connects the continuous transform to the finite DFT, derives its frequency grid and inverse, and makes scaling, symmetry, convolution, Parseval's identity, and multidimensional transforms operational rather than mnemonic.

A metric turns approximation into a precise notion of distance. This chapter separates topology from metric-dependent completeness, develops Cauchy reasoning, and constructs the canonical completion from equivalence classes of Cauchy sequences.

Compactness converts infinite behavior into finite control, while contraction mappings convert a complete metric space into a convergent iteration with an error bound. This chapter explains both mechanisms and the infinite-dimensional failure of bounded-equals-compact.

A norm measures the size of vectors, functions, and sequences; a Banach space is complete in that norm. This chapter develops the core examples, convergence of vector series, equivalent norms, closed subspaces, and the role completeness plays in analysis.

The spaces ℓᵖ and Lᵖ turn sequences and measurable functions into Banach spaces. This chapter explains their norms, Hölder and Minkowski inequalities, density, duality, reflexivity, and the measure-dependent inclusion rules that prevent many common mistakes.

Closed and bounded sets cease to be compact in infinite dimensions. This chapter explains the missing controls—equicontinuity, tails, and translations—through Arzelà–Ascoli and Kolmogorov–Riesz, with counterexamples that expose each hypothesis.

Hilbert spaces are complete inner-product spaces, so Euclidean geometry survives in infinite dimensions. This chapter derives orthogonal projection, normal equations, best approximation, orthogonal complements, and the Riesz representation of continuous functionals.

Orthonormal bases turn Hilbert geometry into stable coordinates. This chapter derives Bessel and Parseval identities, explains completeness and separability, connects Fourier series to abstract expansions, and separates exact Gram–Schmidt from stable numerical orthogonalization.

A linear map between normed spaces is continuous exactly when it is bounded. This chapter develops the operator norm, composition, kernels and ranges, standard examples, Neumann-series inversion, and the distinction between finite-dimensional intuition and infinite-dimensional behavior.

Differential and multiplication operators are often necessarily unbounded, so their domain becomes part of the mathematical object. This chapter explains graphs, closedness, closability, graph norms, dense domains, and why the closed graph theorem does not make every closed operator bounded.

Three Baire-category theorems turn pointwise or qualitative information into uniform operator control. This chapter states their exact hypotheses, shows how they connect, and diagnoses the incomplete-space counterexamples that make those hypotheses necessary.

The continuous dual records every bounded linear measurement of a normed space. Hahn–Banach extends such measurements without increasing their norm and yields norming functionals, point separation, annihilators, and geometric separation.

Weak topologies retain the observations made by continuous functionals while discarding norm detail. This chapter explains weak and weak-* convergence, Banach–Alaoglu compactness, reflexivity, lower semicontinuity, and the sequential caveats hidden by finite-dimensional intuition.

Every bounded operator induces a dual operator, while Hilbert geometry represents that dual action as an adjoint on vectors. This chapter separates the two constructions and develops kernels, ranges, self-adjointness, unitarity, positivity, integral kernels, and domain issues for unbounded adjoints.

The spectrum records where an operator minus a scalar multiple of the identity cannot be stably inverted. This chapter develops resolvent identities, spectral types, Neumann bounds, the spectral-radius formula, and the special geometry of self-adjoint and normal operators.

Compact operators send bounded sets into relatively compact ones, making infinite-dimensional problems behave spectrally like matrices away from zero. This chapter develops compactness tests, the compact spectral theorem, and the Fredholm alternative with its exact solvability condition.

Linear integral equations place the unknown under an integral operator. This chapter classifies Fredholm and Volterra equations, derives Neumann and resolvent kernels, reduces degenerate kernels to matrices, applies the Fredholm alternative, and explains why first-kind inversion is ill-conditioned.

Probability is a disciplined way to describe uncertainty before an outcome is known; a 65% forecast is neither certainty nor a claim that this particular event must happen.

A probability distribution is a complete model for how probability is assigned to a random variable's possible values. PMFs, densities, CDFs, quantiles, transformations, and distribution families are different views of that model, not interchangeable curve-fitting vocabulary.

Conditional probability updates uncertainty after evidence arrives; Bayes' rule combines evidence strength with the base rate that existed before the evidence.

Expectation describes a probability-weighted center, while variance describes how widely outcomes can scatter around it; profitable expectation and a smooth path are different claims.

The law of large numbers explains why averages stabilize, while the central limit theorem approximates their sampling fluctuation; neither removes model error or dependence.

A Markov chain evolves by repeatedly applying a transition kernel: the next-state distribution depends on the current state, not the full recorded path. Stationarity, detailed balance, irreducibility, recurrence, periodicity, and mixing answer different questions, so preserving a target distribution is not by itself evidence that finite samples represent it.

Estimation uses finite data to infer an unknown quantity, while a confidence interval quantifies sampling uncertainty under a declared model rather than the probability that one fixed parameter moves around.

A hypothesis test asks whether observed data are unusually incompatible with a specified null model; it does not measure effect value, truth probability, or trading profitability by itself.

ANOVA turns a question about several group means into a comparison of two variance estimates: variation explained by group membership and variation left within groups. The F-ratio is meaningful only with the design, assumptions, and planned comparisons that make those estimates comparable.

Goodness-of-fit tests measure a particular discrepancy between observed data and a fully stated distributional model. Chi-square compares binned counts; Kolmogorov–Smirnov compares cumulative distributions, and each has failure modes that matter more than the convenience of a p-value.

Experimental design creates comparisons in which an outcome difference can be attributed to controlled factors rather than time, selection, or nuisance variation. Randomization protects causal attribution, replication estimates variability, blocking removes known nuisance variation, and factorial designs expose interactions.

A measurement result is an estimate of a defined measurand accompanied by uncertainty, not a hidden true value plus a known error. A defensible uncertainty budget models every important input, propagates covariance through the measurement equation, applies justified corrections, and preserves metrological traceability.

Dependence means variables carry information about one another; covariance and correlation summarize only part of that relationship and can hide nonlinear or common-event risk.

Calibration checks whether quoted probabilities match observed frequencies, while the Brier score grades each probability against its binary outcome without collapsing it to a class label.

Accuracy grades class labels, not probability quality or economic value; a rational action threshold comes from payoffs and costs rather than automatically from 50%.

PCA finds orthogonal directions that reconstruct observed variation efficiently; factor analysis posits latent causes plus variable-specific noise. Their matrices can look similar, but their objectives, assumptions, non-identifiability, and interpretations are fundamentally different.

Discriminant analysis models class-conditional multivariate distributions and classifies by posterior probability; canonical discriminant directions project data to maximize between-class separation relative to within-class variation. Shared versus class-specific covariance is the boundary between LDA and QDA.

Linear programming optimizes a linear objective over linear constraints. Its power comes from the geometry of convex polyhedra, dual certificates of bounds and infeasibility, and mature sparse solvers—not from memorizing one simplex-table convention.

Entropy measures average uncertainty in a probability distribution; conditional entropy measures what remains after observation; mutual information measures the expected reduction. These quantities concern statistical dependence and coding limits, not meaning, truth or usefulness by themselves.