OCME · Bilingual mathematics & derivations
First Applied Foundations: Dynamics, ODEs, Numerical Error, Optimization, Physics, Biology, and Finance
On this page
- Set
- Function mapping
- Natural number
- Real derivatives at a point and finite difference quotients
- Equation
- Real number
- Absolute value
- Ratio and proportion
- Usual order on the real numbers
- Arithmetic operations
- Sequence
- Natural-number powers and principal square roots
- Discrete-time dynamical system
- Ordinary differential equation
- Numerical approximation and error
- Optimization problem
- Dimensional analysis
- Discrete exponential growth model
- Compound-interest model
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.
Set
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Natural number
Existing canonical prerequisite for the new objects in this pulse.
Real derivatives at a point and finite difference quotients
Existing canonical prerequisite for the new objects in this pulse.
Equation
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Absolute value
Existing canonical prerequisite for the new objects in this pulse.
Ratio and proportion
Existing canonical prerequisite for the new objects in this pulse.
Usual order on the real numbers
Existing canonical prerequisite for the new objects in this pulse.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Natural-number powers and principal square roots
Existing canonical prerequisite for the new objects in this pulse.
Discrete-time dynamical system
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.
Ordinary differential equation
An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems.
Numerical approximation and error
Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value.
Optimization problem
An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects.
Dimensional analysis
Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides.
Discrete exponential growth model
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.
Compound-interest model
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.
Path source & status
path-pulse-foundations-20261020 · candidate