OCME · Bilingual mathematics & derivations
Second Modern Structures: Complex Invariants, Spans, Group Homomorphisms, Convergence, Measure, Processes and Multivariable Functions
On this page
- Complex number
- Absolute value
- Natural-number powers and principal square roots
- Real vector space
- Linear systems in two variables and degenerate constraints
- Finite group
- Subset
- Function mapping
- Metric space
- Sequence
- Quantifiers and explicit domains
- Set
- Set operations
- Real number
- Finite discrete probability distribution
- Real random variables on finite sample spaces
- Plane vectors in a fixed basis
- Coordinate plane
- Complex modulus and conjugation
- Linear combinations and span
- Subgroups and group homomorphisms
- Convergence in a metric space
- Sigma-algebras and measure spaces
- Discrete-time stochastic process
- Multivariable real-valued function
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.
Complex number
Existing canonical prerequisite for the new objects in this pulse.
Absolute value
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.
Real vector space
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.
Finite group
Existing canonical prerequisite for the new objects in this pulse.
Subset
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Metric space
Existing canonical prerequisite for the new objects in this pulse.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Set
Existing canonical prerequisite for the new objects in this pulse.
Set operations
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Finite discrete probability distribution
Existing canonical prerequisite for the new objects in this pulse.
Real random variables on finite sample spaces
Existing canonical prerequisite for the new objects in this pulse.
Plane vectors in a fixed basis
Existing canonical prerequisite for the new objects in this pulse.
Coordinate plane
Existing canonical prerequisite for the new objects in this pulse.
Complex modulus and conjugation
Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares.
Linear combinations and span
A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination.
Subgroups and group homomorphisms
A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table.
Convergence in a metric space
A sequence in a metric space converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x.
Sigma-algebras and measure spaces
A sigma-algebra contains the empty set and is closed under complements and countable unions; a measure is nonnegative, zero on the empty set and countably additive.
Discrete-time stochastic process
A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law.
Multivariable real-valued function
A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data.
Path source & status
path-pulse-foundations-20261023 · candidate