# Third Modern Structures: Bases, Measurability, Markov Chains, Partials, PDEs and Convex Geometry Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Linear combinations and span](/en/mko/mko-linear-combination-span/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-combination-span) [Concept guides](/en/derivations/?q=mko-linear-combination-span) - Stage 1 · required ## [Linear systems in two variables and degenerate constraints](/en/mko/mko-system-linear-equations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-system-linear-equations) [Concept guides](/en/derivations/?q=mko-system-linear-equations) - Stage 1 · required ## [Determinants of real 2×2 matrices](/en/mko/mko-determinant/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-determinant) [Concept guides](/en/derivations/?q=mko-determinant) - Stage 1 · required ## [Sigma-algebras and measure spaces](/en/mko/mko-sigma-algebra-measure-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sigma-algebra-measure-space) [Concept guides](/en/derivations/?q=mko-sigma-algebra-measure-space) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-operations) [Concept guides](/en/derivations/?q=mko-set-operations) - Stage 1 · required ## [Discrete-time stochastic process](/en/mko/mko-discrete-stochastic-process/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-discrete-stochastic-process) [Concept guides](/en/derivations/?q=mko-discrete-stochastic-process) - Stage 1 · required ## [Finite matrices: dimensions, indexing, and transpose](/en/mko/mko-matrix/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-matrix) [Concept guides](/en/derivations/?q=mko-matrix) - Stage 1 · required ## [Conditional probability in finite models](/en/mko/mko-conditional-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-conditional-probability) [Concept guides](/en/derivations/?q=mko-conditional-probability) - Stage 1 · required ## [Multivariable real-valued function](/en/mko/mko-multivariable-real-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-multivariable-real-function) [Concept guides](/en/derivations/?q=mko-multivariable-real-function) - Stage 1 · required ## [Real derivatives at a point and finite difference quotients](/en/mko/mko-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-derivative) [Concept guides](/en/derivations/?q=mko-derivative) - Stage 1 · required ## [Average rate of change between distinct inputs](/en/mko/mko-rate-of-change/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-rate-of-change) [Concept guides](/en/derivations/?q=mko-rate-of-change) - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-vector-space) [Concept guides](/en/derivations/?q=mko-real-vector-space) - Stage 1 · required ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-ratio-proportion) [Concept guides](/en/derivations/?q=mko-ratio-proportion) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-equation) [Concept guides](/en/derivations/?q=mko-equation) - Stage 1 · required ## [Differentiating polynomial sums, products, and compositions](/en/mko/mko-derivative-rules/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-derivative-rules) [Concept guides](/en/derivations/?q=mko-derivative-rules) - Stage 1 · required ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subset) [Concept guides](/en/derivations/?q=mko-subset) - Stage 2 · optional ## [Linear independence and bases](/en/mko/mko-linear-independence-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space. [Related questions](/en/explore/?mko=mko-linear-independence-basis) [Concept guides](/en/derivations/?q=mko-linear-independence-basis) - Stage 3 · optional ## [Measurable function](/en/mko/mko-measurable-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra. [Related questions](/en/explore/?mko=mko-measurable-function) [Concept guides](/en/derivations/?q=mko-measurable-function) - Stage 4 · optional ## [Finite-state Markov chain](/en/mko/mko-finite-markov-chain/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities. [Related questions](/en/explore/?mko=mko-finite-markov-chain) [Concept guides](/en/derivations/?q=mko-finite-markov-chain) - Stage 5 · optional ## [Partial derivative](/en/mko/mko-partial-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero. [Related questions](/en/explore/?mko=mko-partial-derivative) [Concept guides](/en/derivations/?q=mko-partial-derivative) - Stage 6 · optional ## [Finite convex combination](/en/mko/mko-finite-convex-combination/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one. [Related questions](/en/explore/?mko=mko-finite-convex-combination) [Concept guides](/en/derivations/?q=mko-finite-convex-combination) - Stage 7 · optional ## [Partial differential equation](/en/mko/mko-partial-differential-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention. [Related questions](/en/explore/?mko=mko-partial-differential-equation) [Concept guides](/en/derivations/?q=mko-partial-differential-equation) - Stage 8 · optional ## [Convex set](/en/mko/mko-convex-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261024%2F) A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set. [Related questions](/en/explore/?mko=mko-convex-set) [Concept guides](/en/derivations/?q=mko-convex-set) Path source & statuspath-pulse-foundations-20261024 · candidate [Original paths JSON](/data/architecture/learning-paths.json)