# Second Applied Foundations: Representation, Assessment Records, Periodicity, Initial Values, Bisection, Linear Programming and Capacity-Limited Growth 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 ## [Variables and algebraic expressions](/en/mko/mko-variable-expression/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-variable-expression) [Concept guides](/en/derivations/?q=mko-variable-expression) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%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 ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 1 · required ## [Proposition](/en/mko/mko-proposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-proposition) [Concept guides](/en/derivations/?q=mko-proposition) - Stage 1 · required ## [Counterexamples and domain eligibility](/en/mko/mko-counterexample/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-counterexample) [Concept guides](/en/derivations/?q=mko-counterexample) - Stage 1 · required ## [Discrete-time dynamical system](/en/mko/mko-discrete-dynamical-system/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-discrete-dynamical-system) [Concept guides](/en/derivations/?q=mko-discrete-dynamical-system) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%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 ## [Ordinary differential equation](/en/mko/mko-ordinary-differential-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-ordinary-differential-equation) [Concept guides](/en/derivations/?q=mko-ordinary-differential-equation) - Stage 1 · required ## [Domain, codomain, and range](/en/mko/mko-domain-codomain-range/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%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 ## [Numerical approximation and error](/en/mko/mko-numerical-approximation-error/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-numerical-approximation-error) [Concept guides](/en/derivations/?q=mko-numerical-approximation-error) - Stage 1 · required ## [Point values and continuity of real functions](/en/mko/mko-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-continuity) [Concept guides](/en/derivations/?q=mko-continuity) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%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-20261021%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 ## [Linear equation in one variable](/en/mko/mko-linear-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-equation) [Concept guides](/en/derivations/?q=mko-linear-equation) - Stage 1 · required ## [Linear systems in two variables and degenerate constraints](/en/mko/mko-system-linear-equations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%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 ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 1 · required ## [Discrete exponential growth model](/en/mko/mko-discrete-exponential-growth/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-discrete-exponential-growth) [Concept guides](/en/derivations/?q=mko-discrete-exponential-growth) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%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 2 · optional ## [Mathematical representation](/en/mko/mko-mathematical-representation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Mathematical representation separately records verbal rules, symbolic expressions, finite tables, ordered pairs and coordinate data for the same object, using explicit semantic bridges to check agreement. [Related questions](/en/explore/?mko=mko-mathematical-representation) [Concept guides](/en/derivations/?q=mko-mathematical-representation) - Stage 3 · optional ## [Diagnostic mathematical assessment records](/en/mko/mko-diagnostic-mathematical-assessment/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) A diagnostic mathematical assessment record binds an item, its expected result, a response and an explicit judgment rule, producing only a traceable rule-agreement status. [Related questions](/en/explore/?mko=mko-diagnostic-mathematical-assessment) [Concept guides](/en/derivations/?q=mko-diagnostic-mathematical-assessment) - Stage 4 · optional ## [Orbits, fixed points and periodic points](/en/mko/mko-orbit-fixed-periodic/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time. [Related questions](/en/explore/?mko=mko-orbit-fixed-periodic) [Concept guides](/en/derivations/?q=mko-orbit-fixed-periodic) - Stage 5 · optional ## [Initial-value problem](/en/mko/mko-initial-value-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness. [Related questions](/en/explore/?mko=mko-initial-value-problem) [Concept guides](/en/derivations/?q=mko-initial-value-problem) - Stage 6 · optional ## [Finite sign-bracket updates in the bisection method](/en/mko/mko-bisection-method/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width. [Related questions](/en/explore/?mko=mko-bisection-method) [Concept guides](/en/derivations/?q=mko-bisection-method) - Stage 7 · optional ## [Linear programming](/en/mko/mko-linear-programming/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) Linear programming optimizes a linear objective over a feasible domain defined by linear equalities or inequalities; this exercise slice checks only a complete explicit finite candidate grid. [Related questions](/en/explore/?mko=mko-linear-programming) [Concept guides](/en/derivations/?q=mko-linear-programming) - Stage 8 · optional ## [Discrete logistic increment model](/en/mko/mko-discrete-logistic-growth/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261021%2F) For fixed capacity K and 0≤r≤1, this entry uses x_(n+1)=x_n+r*x_n*(1−x_n/K) to describe each capacity-limited discrete increment. [Related questions](/en/explore/?mko=mko-discrete-logistic-growth) [Concept guides](/en/derivations/?q=mko-discrete-logistic-growth) Path source & statuspath-pulse-foundations-20261021 · candidate [Original paths JSON](/data/architecture/learning-paths.json)