OCME · Bilingual mathematics & derivations
Wave6 Sixth Layer: Cauchy, Taylor, Compact Closed Subsets, Improper Integrals, Convergence Radii, Legendre and Subset Construction
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- Lagrange theorem for finite groups
- Prime natural number
- Finite group
- Finite group action
- Differentiating polynomial sums, products, and compositions
- Univariate polynomials over the rationals
- Factorial
- Higher-order derivative
- Hausdorff space
- Compact subset
- Open and closed sets
- Signed polynomial definite integrals on finite intervals
- Function limits on punctured real domains
- Sequence limits and exact tail cutoffs
- Absolute convergence of series
- Infinite series
- Supremum and infimum
- Absolute value
- Natural-number powers and principal square roots
- Quadratic residue
- Function mapping
- Nondeterministic finite automaton
- Deterministic finite automaton
- Subset
- Cauchy theorem for finite groups
- Taylor polynomial
- A compact subset of a Hausdorff space is closed
- Improper integral
- Radius of convergence of a power series
- Legendre symbol
- Subset construction from an NFA to a DFA
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.
Lagrange theorem for finite groups
Existing canonical prerequisite for the new objects in this pulse.
Prime natural number
Existing canonical prerequisite for the new objects in this pulse.
Finite group
Existing canonical prerequisite for the new objects in this pulse.
Finite group action
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.
Univariate polynomials over the rationals
Existing canonical prerequisite for the new objects in this pulse.
Factorial
Existing canonical prerequisite for the new objects in this pulse.
Higher-order derivative
Existing canonical prerequisite for the new objects in this pulse.
Hausdorff space
Existing canonical prerequisite for the new objects in this pulse.
Compact subset
Existing canonical prerequisite for the new objects in this pulse.
Open and closed sets
Existing canonical prerequisite for the new objects in this pulse.
Signed polynomial definite integrals on finite intervals
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.
Sequence limits and exact tail cutoffs
Existing canonical prerequisite for the new objects in this pulse.
Absolute convergence of series
Existing canonical prerequisite for the new objects in this pulse.
Infinite series
Existing canonical prerequisite for the new objects in this pulse.
Supremum and infimum
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.
Quadratic residue
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Nondeterministic finite automaton
Existing canonical prerequisite for the new objects in this pulse.
Deterministic finite automaton
Existing canonical prerequisite for the new objects in this pulse.
Subset
Existing canonical prerequisite for the new objects in this pulse.
Cauchy theorem for finite groups
If a prime p divides the order of a finite group G, G contains an element of exact order p.
Taylor polynomial
At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial.
A compact subset of a Hausdorff space is closed
If K is compact in the Hausdorff space X, then K is closed in X.
Improper integral
Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently.
Radius of convergence of a power series
Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions.
Legendre symbol
For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively.
Subset construction from an NFA to a DFA
Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language.
Path source & status
path-pulse-foundations-20261112 · candidate