# First Applied Foundations: Dynamics, ODEs, Numerical Error, Optimization, Physics, Biology, and Finance 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 ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%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 ## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number) - Stage 1 · required ## [Real derivatives at a point and finite difference quotients](/en/mko/mko-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%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 ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%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 ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) - Stage 1 · required ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%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 ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%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 ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%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 ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%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 ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 2 · optional ## [Discrete-time dynamical system](/en/mko/mko-discrete-dynamical-system/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability. [Related questions](/en/explore/?mko=mko-discrete-dynamical-system) [Concept guides](/en/derivations/?q=mko-discrete-dynamical-system) - Stage 3 · optional ## [Ordinary differential equation](/en/mko/mko-ordinary-differential-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems. [Related questions](/en/explore/?mko=mko-ordinary-differential-equation) [Concept guides](/en/derivations/?q=mko-ordinary-differential-equation) - Stage 4 · optional ## [Numerical approximation and error](/en/mko/mko-numerical-approximation-error/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value. [Related questions](/en/explore/?mko=mko-numerical-approximation-error) [Concept guides](/en/derivations/?q=mko-numerical-approximation-error) - Stage 5 · optional ## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects. [Related questions](/en/explore/?mko=mko-optimization-problem) [Concept guides](/en/derivations/?q=mko-optimization-problem) - Stage 6 · optional ## [Dimensional analysis](/en/mko/mko-dimensional-analysis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides. [Related questions](/en/explore/?mko=mko-dimensional-analysis) [Concept guides](/en/derivations/?q=mko-dimensional-analysis) - Stage 7 · optional ## [Discrete exponential growth model](/en/mko/mko-discrete-exponential-growth/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n. [Related questions](/en/explore/?mko=mko-discrete-exponential-growth) [Concept guides](/en/derivations/?q=mko-discrete-exponential-growth) - Stage 8 · optional ## [Compound-interest model](/en/mko/mko-compound-interest/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261020%2F) With a fixed rate i per period and no additional cash flows, balances satisfy B_(n+1)=(1+i)B_n and hence B_n=B_0(1+i)^n. [Related questions](/en/explore/?mko=mko-compound-interest) [Concept guides](/en/derivations/?q=mko-compound-interest) Path source & statuspath-pulse-foundations-20261020 · candidate [Original paths JSON](/data/architecture/learning-paths.json)