# Wave6 Ninth Layer: First Isomorphism, Centers, Bias-Variance, LP Duality, Max-Flow Min-Cut, Minimum Covers and Many-One Reduction 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. - Stage 1 · required ## [Kernel of a finite group homomorphism](/en/mko/mko-finite-group-homomorphism-kernel/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group-homomorphism-kernel) [Concept guides](/en/derivations/?q=mko-finite-group-homomorphism-kernel) - Stage 1 · required ## [Finite quotient group](/en/mko/mko-finite-quotient-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-quotient-group) [Concept guides](/en/derivations/?q=mko-finite-quotient-group) - Stage 1 · required ## [Subgroups and group homomorphisms](/en/mko/mko-subgroup-homomorphism/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subgroup-homomorphism) [Concept guides](/en/derivations/?q=mko-subgroup-homomorphism) - Stage 1 · required ## [Bijectivity and unique preimages](/en/mko/mko-bijective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-bijective) [Concept guides](/en/derivations/?q=mko-bijective) - Stage 1 · required ## [Finite group](/en/mko/mko-finite-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group) [Concept guides](/en/derivations/?q=mko-finite-group) - Stage 1 · required ## [Conjugacy class in a finite group](/en/mko/mko-finite-group-conjugacy-class/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group-conjugacy-class) [Concept guides](/en/derivations/?q=mko-finite-group-conjugacy-class) - Stage 1 · required ## [Finite normal subgroup](/en/mko/mko-finite-normal-subgroup/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-normal-subgroup) [Concept guides](/en/derivations/?q=mko-finite-normal-subgroup) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Mean squared error of an estimator](/en/mko/mko-estimator-mean-squared-error/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-estimator-mean-squared-error) [Concept guides](/en/derivations/?q=mko-estimator-mean-squared-error) - Stage 1 · required ## [Estimator bias](/en/mko/mko-estimator-bias/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-estimator-bias) [Concept guides](/en/derivations/?q=mko-estimator-bias) - Stage 1 · required ## [Estimator variance](/en/mko/mko-estimator-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-estimator-variance) [Concept guides](/en/derivations/?q=mko-estimator-variance) - Stage 1 · required ## [Standard form of a linear program](/en/mko/mko-linear-program-standard-form/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-program-standard-form) [Concept guides](/en/derivations/?q=mko-linear-program-standard-form) - Stage 1 · required ## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-optimization-problem) [Concept guides](/en/derivations/?q=mko-optimization-problem) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 1 · required ## [Finite flow network](/en/mko/mko-finite-flow-network/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-flow-network) [Concept guides](/en/derivations/?q=mko-finite-flow-network) - Stage 1 · required ## [Linear programming](/en/mko/mko-linear-programming/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-programming) [Concept guides](/en/derivations/?q=mko-linear-programming) - Stage 1 · required ## [Vertex cover](/en/mko/mko-vertex-cover/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vertex-cover) [Concept guides](/en/derivations/?q=mko-vertex-cover) - Stage 1 · required ## [Cardinality comparison of sets](/en/mko/mko-cardinality-comparison/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-cardinality-comparison) [Concept guides](/en/derivations/?q=mko-cardinality-comparison) - Stage 1 · required ## [Decision problem](/en/mko/mko-decision-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-decision-problem) [Concept guides](/en/derivations/?q=mko-decision-problem) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) - Stage 1 · required ## [Turing-machine decision model](/en/mko/mko-turing-machine-decision-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-turing-machine-decision-model) [Concept guides](/en/derivations/?q=mko-turing-machine-decision-model) - Stage 2 · optional ## [First isomorphism theorem for finite groups](/en/mko/mko-first-isomorphism-theorem-finite-groups/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain. [Related questions](/en/explore/?mko=mko-first-isomorphism-theorem-finite-groups) [Concept guides](/en/derivations/?q=mko-first-isomorphism-theorem-finite-groups) - Stage 3 · optional ## [Center of a finite group](/en/mko/mko-finite-group-center/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) The center consists of all elements commuting with every group element and is a normal subgroup. [Related questions](/en/explore/?mko=mko-finite-group-center) [Concept guides](/en/derivations/?q=mko-finite-group-center) - Stage 4 · optional ## [Bias-variance decomposition](/en/mko/mko-bias-variance-decomposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias. [Related questions](/en/explore/?mko=mko-bias-variance-decomposition) [Concept guides](/en/derivations/?q=mko-bias-variance-decomposition) - Stage 5 · optional ## [Linear-programming duality](/en/mko/mko-linear-program-duality/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) For primal maximize c^T*x with A*x≤b,x≥0, the dual minimizes b^T*y with A^T*y≥c,y≥0; feasible pairs satisfy weak duality. [Related questions](/en/explore/?mko=mko-linear-program-duality) [Concept guides](/en/derivations/?q=mko-linear-program-duality) - Stage 6 · optional ## [Max-flow min-cut theorem](/en/mko/mko-max-flow-min-cut-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts. [Related questions](/en/explore/?mko=mko-max-flow-min-cut-theorem) [Concept guides](/en/derivations/?q=mko-max-flow-min-cut-theorem) - Stage 7 · optional ## [Minimum vertex cover](/en/mko/mko-minimum-vertex-cover/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph. [Related questions](/en/explore/?mko=mko-minimum-vertex-cover) [Concept guides](/en/derivations/?q=mko-minimum-vertex-cover) - Stage 8 · optional ## [Many-one reduction](/en/mko/mko-many-one-reduction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261115%2F) A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x. [Related questions](/en/explore/?mko=mko-many-one-reduction) [Concept guides](/en/derivations/?q=mko-many-one-reduction) Path source & statuspath-pulse-foundations-20261115 · candidate [Original paths JSON](/data/architecture/learning-paths.json)