OCME · Bilingual mathematics & derivations
Wave6 Seventh Layer: Normal Subgroups, Real Exponential, Analytic OGFs, Euler Criterion, Turing Decisions, Gradients and Flow Networks
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- Finite group
- Subgroups and group homomorphisms
- Quantifiers and explicit domains
- Infinite series
- Real number
- Function mapping
- Sequence limits and exact tail cutoffs
- Radius of convergence of a power series
- Factorial
- Absolute convergence of series
- Binomial theorem
- Recurrence relation
- Sequence
- Euler's theorem in number theory
- Quadratic residue
- Legendre symbol
- Abstract computational model
- Decision problem
- Deterministic finite automaton
- Algorithm correctness
- Partial derivative
- Plane vectors in a fixed basis
- Multivariable real-valued function
- Finite directed graph
- Linear programming
- Finite normal subgroup
- Real exponential function
- Analytic ordinary generating function
- Euler criterion for quadratic residues
- Turing-machine decision model
- Gradient vector
- Finite flow network
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.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Infinite series
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
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.
Radius of convergence of a power series
Existing canonical prerequisite for the new objects in this pulse.
Factorial
Existing canonical prerequisite for the new objects in this pulse.
Absolute convergence of series
Existing canonical prerequisite for the new objects in this pulse.
Binomial theorem
Existing canonical prerequisite for the new objects in this pulse.
Recurrence relation
Existing canonical prerequisite for the new objects in this pulse.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Euler's theorem in number theory
Existing canonical prerequisite for the new objects in this pulse.
Quadratic residue
Existing canonical prerequisite for the new objects in this pulse.
Legendre symbol
Existing canonical prerequisite for the new objects in this pulse.
Abstract computational model
Existing canonical prerequisite for the new objects in this pulse.
Decision problem
Existing canonical prerequisite for the new objects in this pulse.
Deterministic finite automaton
Existing canonical prerequisite for the new objects in this pulse.
Algorithm correctness
Existing canonical prerequisite for the new objects in this pulse.
Partial derivative
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.
Multivariable real-valued function
Existing canonical prerequisite for the new objects in this pulse.
Finite directed graph
Existing canonical prerequisite for the new objects in this pulse.
Linear programming
Existing canonical prerequisite for the new objects in this pulse.
Finite normal subgroup
A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset.
Real exponential function
Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0.
Analytic ordinary generating function
An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain.
Euler criterion for quadratic residues
For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p).
Turing-machine decision model
Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer.
Gradient vector
In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector.
Finite flow network
A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink.
Path source & status
path-pulse-foundations-20261113 · candidate