OCME · Bilingual mathematics & derivations
Third Modern Structures: Bases, Measurability, Markov Chains, Partials, PDEs and Convex Geometry
On this page
- Linear combinations and span
- Linear systems in two variables and degenerate constraints
- Determinants of real 2×2 matrices
- Sigma-algebras and measure spaces
- Function mapping
- Set operations
- Discrete-time stochastic process
- Finite matrices: dimensions, indexing, and transpose
- Conditional probability in finite models
- Multivariable real-valued function
- Real derivatives at a point and finite difference quotients
- Average rate of change between distinct inputs
- Real vector space
- Ratio and proportion
- Arithmetic operations
- Equation
- Differentiating polynomial sums, products, and compositions
- Subset
- Linear independence and bases
- Measurable function
- Finite-state Markov chain
- Partial derivative
- Finite convex combination
- Partial differential equation
- Convex set
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.
Linear combinations and span
Existing canonical prerequisite for the new objects in this pulse.
Linear systems in two variables and degenerate constraints
Existing canonical prerequisite for the new objects in this pulse.
Determinants of real 2×2 matrices
Existing canonical prerequisite for the new objects in this pulse.
Sigma-algebras and measure spaces
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Set operations
Existing canonical prerequisite for the new objects in this pulse.
Discrete-time stochastic process
Existing canonical prerequisite for the new objects in this pulse.
Finite matrices: dimensions, indexing, and transpose
Existing canonical prerequisite for the new objects in this pulse.
Conditional probability in finite models
Existing canonical prerequisite for the new objects in this pulse.
Multivariable real-valued function
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.
Average rate of change between distinct inputs
Existing canonical prerequisite for the new objects in this pulse.
Real vector space
Existing canonical prerequisite for the new objects in this pulse.
Ratio and proportion
Existing canonical prerequisite for the new objects in this pulse.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Equation
Existing canonical prerequisite for the new objects in this pulse.
Differentiating polynomial sums, products, and compositions
Existing canonical prerequisite for the new objects in this pulse.
Subset
Existing canonical prerequisite for the new objects in this pulse.
Linear independence and bases
Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space.
Measurable function
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.
Finite-state Markov chain
A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities.
Partial derivative
A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero.
Finite convex combination
A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one.
Partial differential equation
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.
Convex set
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.
Path source & status
path-pulse-foundations-20261024 · candidate