# Modern Structures Wave3 Closure: Boundary Values, Convex Functions, Convex Optimization and Four Density Extensions 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 ## [Partial differential equation](/en/mko/mko-partial-differential-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-partial-differential-equation) [Concept guides](/en/derivations/?q=mko-partial-differential-equation) - Stage 1 · required ## [Domain, codomain, and range](/en/mko/mko-domain-codomain-range/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-domain-codomain-range) [Concept guides](/en/derivations/?q=mko-domain-codomain-range) - Stage 1 · required ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%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 ## [Convex set](/en/mko/mko-convex-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-convex-set) [Concept guides](/en/derivations/?q=mko-convex-set) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%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 ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 1 · required ## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-optimization-problem) [Concept guides](/en/derivations/?q=mko-optimization-problem) - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%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 ## [Linear combinations and span](/en/mko/mko-linear-combination-span/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%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-20261025%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-20261025%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-20261025%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 ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%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-20261025%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-20261025%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-20261025%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-20261025%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-20261025%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-20261025%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 2 · optional ## [Boundary-value problem](/en/mko/mko-boundary-value-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness. [Related questions](/en/explore/?mko=mko-boundary-value-problem) [Concept guides](/en/derivations/?q=mko-boundary-value-problem) - Stage 2 · optional ## [Convex function](/en/mko/mko-convex-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values. [Related questions](/en/explore/?mko=mko-convex-function) [Concept guides](/en/derivations/?q=mko-convex-function) - Stage 2 · optional ## [Linear independence and bases](/en/mko/mko-linear-independence-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Deepen the existing R² basis semantics through explicit reconstruction and nontrivial zero-sum certificates, separating span, dependence and coordinate uniqueness. [Related questions](/en/explore/?mko=mko-linear-independence-basis) [Concept guides](/en/derivations/?q=mko-linear-independence-basis) - Stage 2 · optional ## [Measurable function](/en/mko/mko-measurable-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Deepen complement and union preimage laws, constant maps and source-sigma refinement on complete finite measurable spaces. [Related questions](/en/explore/?mko=mko-measurable-function) [Concept guides](/en/derivations/?q=mko-measurable-function) - Stage 2 · optional ## [Finite-state Markov chain](/en/mko/mko-finite-markov-chain/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Deepen exact n-step matrices, endpoint joint probabilities and paths conditional on a given initial state in a finite time-homogeneous kernel. [Related questions](/en/explore/?mko=mko-finite-markov-chain) [Concept guides](/en/derivations/?q=mko-finite-markov-chain) - Stage 2 · optional ## [Partial derivative](/en/mko/mko-partial-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Deepen the x/y partial polynomials of a quadratic, fixed-x y-slices and limits after symbolic difference-quotient cancellation. [Related questions](/en/explore/?mko=mko-partial-derivative) [Concept guides](/en/derivations/?q=mko-partial-derivative) - Stage 3 · optional ## [Convex optimization problem](/en/mko/mko-convex-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261025%2F) Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations. [Related questions](/en/explore/?mko=mko-convex-optimization-problem) [Concept guides](/en/derivations/?q=mko-convex-optimization-problem) Path source & statuspath-pulse-foundations-20261025 · candidate [Original paths JSON](/data/architecture/learning-paths.json)