Math Tier 3 — Family Index Catalog
This is the Tier 3 layer of the Math knowledge base: comprehensive family indexes that catalog entire zoos of mathematical objects (distributions, algorithms, kernels, methods, functions). Where Tier 1 ([[Math/_index]]) gives conceptual entry points and Tier 2 gives focused topical notes, Tier 3 is the encyclopedic reference — dense tables, formulas, and cross-relationships designed for lookup, not narrative reading.
Each Tier 3 note targets ~500 lines, lists 30–80 distinct items with their defining formulas, properties, and use cases, and ends with a relationships diagram and decision tree for selection.
Planned Tier 3 family indexes
| Family index | Items catalogued | Status | Cousin (Tier 1/2) |
|---|---|---|---|
[[Math/Tier3/probability-distribution-zoo]] | 60+ probability distributions (discrete + continuous + multivariate + circular + Bayesian non-parametric) with PMF/PDF, moments, MGF, conjugate priors, transformations | drafted 2026-05-17 | [[Math/probability-distributions]] |
[[Math/Tier3/statistical-distributions-catalog-extended]] | 100+ distributions extending the zoo — heavy-tailed for finance, survival, circular/directional, empirical/nonparametric, EVT, tests + diagnostics (Wave 18) | done 2026-05-22 | [[Math/probability-distributions]], [[Math/copulas-and-dependence]] |
[[Math/Tier3/optimization-algorithm-taxonomy]] | LP/QP/SDP/MILP/NLP, SGD + momentum + Adam + L-BFGS, ADMM, metaheuristics, combinatorial, distributed, robust + DRO | [x] done | [[Math/convex-optimization]], [[Math/gradient-descent-variants]] |
[[Math/Tier3/convolutional-kernel-zoo]] | Image filters, smoothing, edge detection, morphological, CNN kernel families | [x] done | [[Math/gaussian-processes]] |
[[Math/Tier3/ode-pde-solver-catalog]] | Runge-Kutta, multistep, FEM, FD, FV, spectral, multigrid, PINNs | [x] done | [[Math/ode-numerical-methods]], [[Math/pde-methods]] |
[[Math/Tier3/numerical-methods-reference]] | Quadrature, interpolation, root-finding, linear solvers, eigenvalues, floating-point pitfalls | [x] done | [[Math/numerical-linear-algebra]] |
[[Math/Tier3/sampling-algorithms-catalog]] | Inverse-CDF, rejection, importance, MCMC (Metropolis-Hastings, Gibbs, HMC, NUTS, SMC), Langevin, normalizing flows | [x] done 2026-05-23 | [[Math/mcmc-sampling]], [[Math/bayesian-inference]] |
[[Math/Tier3/special-functions]] | Gamma, Beta, error function, Bessel, hypergeometric, Legendre, Hermite, Laguerre, Chebyshev, Riemann zeta, polylogarithm, elliptic integrals | planned | — |
[[Math/Tier3/named-inequalities]] | Cauchy-Schwarz, Hölder, Minkowski, Jensen, Markov, Chebyshev, Chernoff, Hoeffding, Talagrand, log-Sobolev, Poincaré, Brascamp-Lieb, Pinsker, data-processing | planned | [[Math/probability-fundamentals]], [[Math/information-theory]] |
[[Math/Tier3/lie-group-catalog]] | SO(2), SO(3), SE(3), SU(2), SL(n), GL(n), O(n), U(n), Sp(2n), Heisenberg, exceptional groups, affine + dilation + similarity, Lie algebras + exponential maps | planned | [[Math/lie-groups-so3-se3]] |
How to use Tier 3
- Lookup mode: scan the tables; jump to the row, copy the formula, check the parameter relationship.
- Selection mode: read the decision tree at the bottom of each note — answers “given this problem, which member of the family fits?”
- Relationships mode: read the family-tree diagram — answers “what is this a special case of? what does it limit to? what is its conjugate / dual?”
Treat Tier 3 as a time-stamped reference, not a textbook. Verify formulas against an authoritative source (DLMF, Abramowitz & Stegun, Casella & Berger, Bishop, Murphy) before deploying in production code.
Cross-links
[[Math/_index]]— Tier 1 conceptual map[[System/Knowledge-Graph]]— overall vault MOC[[Math/probability-fundamentals]],[[Math/probability-distributions]]— Tier 1/2 cousins for the probability zoo[[Math/convex-optimization]],[[Math/gradient-descent-variants]]— Tier 2 cousins for the optimization taxonomy[[Math/gaussian-processes]]— Tier 2 cousin for the kernel zoo
— Last updated 2026-05-24.