OCME · Bilingual mathematics & derivations
Wave6 Fifth Layer: Lagrange, Higher Derivatives, Hausdorff Spaces, Continuous CDFs, Catalan Numbers, Quadratic Residues and NFAs
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- Finite group
- Subgroups and group homomorphisms
- Natural-number divisibility
- Cardinality comparison of sets
- Quotient set
- Real derivatives at a point and finite difference quotients
- Differentiating polynomial sums, products, and compositions
- Mathematical induction
- Topological space
- Neighborhood
- Quantifiers and explicit domains
- Cumulative distribution function
- Point values and continuity of real functions
- Function limits on punctured real domains
- Usual order on the real numbers
- Sigma-algebras and measure spaces
- Recurrence relation
- Binomial theorem
- Basic counting principles
- Natural-number congruence modulo a positive modulus
- Prime natural number
- Natural-number powers and principal square roots
- Deterministic finite automaton
- Binary relations and set projections
- Subset
- Lagrange theorem for finite groups
- Higher-order derivative
- Hausdorff space
- Continuous real distribution function
- Catalan numbers
- Quadratic residue
- Nondeterministic finite automaton
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.
Finite group
Existing canonical prerequisite for the new objects in this pulse.
Subgroups and group homomorphisms
Existing canonical prerequisite for the new objects in this pulse.
Natural-number divisibility
Existing canonical prerequisite for the new objects in this pulse.
Cardinality comparison of sets
Existing canonical prerequisite for the new objects in this pulse.
Quotient set
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.
Differentiating polynomial sums, products, and compositions
Existing canonical prerequisite for the new objects in this pulse.
Mathematical induction
Existing canonical prerequisite for the new objects in this pulse.
Topological space
Existing canonical prerequisite for the new objects in this pulse.
Neighborhood
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Cumulative distribution function
Existing canonical prerequisite for the new objects in this pulse.
Point values and continuity of real functions
Existing canonical prerequisite for the new objects in this pulse.
Function limits on punctured real domains
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.
Sigma-algebras and measure spaces
Existing canonical prerequisite for the new objects in this pulse.
Recurrence relation
Existing canonical prerequisite for the new objects in this pulse.
Binomial theorem
Existing canonical prerequisite for the new objects in this pulse.
Basic counting principles
Existing canonical prerequisite for the new objects in this pulse.
Natural-number congruence modulo a positive modulus
Existing canonical prerequisite for the new objects in this pulse.
Prime natural number
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.
Deterministic finite automaton
Existing canonical prerequisite for the new objects in this pulse.
Binary relations and set projections
Existing canonical prerequisite for the new objects in this pulse.
Subset
Existing canonical prerequisite for the new objects in this pulse.
Lagrange theorem for finite groups
Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|.
Higher-order derivative
Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'.
Hausdorff space
A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods.
Continuous real distribution function
Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1.
Catalan numbers
With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence.
Quadratic residue
Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p.
Nondeterministic finite automaton
An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance.
Path source & status
path-pulse-foundations-20261111 · candidate