OCME · Bilingual mathematics & derivations
Wave5 Fifth Layer: Direct Sums, Duals, Euclid, Euler Trails, Homeomorphisms, Sampling Laws and the WLLN
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- Real vector subspace
- Set operations
- Linear independence and bases
- Linear map between real vector spaces
- Real vector space
- Real number
- Greatest common divisor and coprimality
- Natural-number divisibility
- Algorithm correctness
- Graph walks and simple paths
- Finite simple undirected graphs
- Connected component
- Continuous maps between topological spaces
- Bijectivity and unique preimages
- Inverse functions of bijections
- Topological space
- Parametric statistical model
- Statistical estimator
- Finite discrete probability distribution
- Independent identically distributed sequence
- Expectation in finite models
- Variance in finite probability models
- Convergence in probability
- Direct sum of subspaces
- Dual vector space
- Euclidean algorithm
- Euler trails and circuits
- Homeomorphism
- Sampling distribution
- Weak law of large numbers
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.
Real vector subspace
Existing canonical prerequisite for the new objects in this pulse.
Set operations
Existing canonical prerequisite for the new objects in this pulse.
Linear independence and bases
Existing canonical prerequisite for the new objects in this pulse.
Linear map between real vector spaces
Existing canonical prerequisite for the new objects in this pulse.
Real vector space
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Greatest common divisor and coprimality
Existing canonical prerequisite for the new objects in this pulse.
Natural-number divisibility
Existing canonical prerequisite for the new objects in this pulse.
Algorithm correctness
Existing canonical prerequisite for the new objects in this pulse.
Graph walks and simple paths
Existing canonical prerequisite for the new objects in this pulse.
Finite simple undirected graphs
Existing canonical prerequisite for the new objects in this pulse.
Connected component
Existing canonical prerequisite for the new objects in this pulse.
Continuous maps between topological spaces
Existing canonical prerequisite for the new objects in this pulse.
Bijectivity and unique preimages
Existing canonical prerequisite for the new objects in this pulse.
Inverse functions of bijections
Existing canonical prerequisite for the new objects in this pulse.
Topological space
Existing canonical prerequisite for the new objects in this pulse.
Parametric statistical model
Existing canonical prerequisite for the new objects in this pulse.
Statistical estimator
Existing canonical prerequisite for the new objects in this pulse.
Finite discrete probability distribution
Existing canonical prerequisite for the new objects in this pulse.
Independent identically distributed sequence
Existing canonical prerequisite for the new objects in this pulse.
Expectation in finite models
Existing canonical prerequisite for the new objects in this pulse.
Variance in finite probability models
Existing canonical prerequisite for the new objects in this pulse.
Convergence in probability
Existing canonical prerequisite for the new objects in this pulse.
Direct sum of subspaces
The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation.
Dual vector space
The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis.
Euclidean algorithm
The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders.
Euler trails and circuits
An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail.
Homeomorphism
A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism.
Sampling distribution
At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws.
Weak law of large numbers
For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean.
Path source & status
path-pulse-foundations-20261105 · candidate