# Second Modern Structures: Complex Invariants, Spans, Group Homomorphisms, Convergence, Measure, Processes and Multivariable Functions 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 ## [Complex number](/en/mko/mko-complex-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-complex-number) [Concept guides](/en/derivations/?q=mko-complex-number) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) - Stage 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-vector-space) [Concept guides](/en/derivations/?q=mko-real-vector-space) - Stage 1 · required ## [Linear systems in two variables and degenerate constraints](/en/mko/mko-system-linear-equations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-system-linear-equations) [Concept guides](/en/derivations/?q=mko-system-linear-equations) - Stage 1 · required ## [Finite group](/en/mko/mko-finite-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%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 ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subset) [Concept guides](/en/derivations/?q=mko-subset) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%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 ## [Metric space](/en/mko/mko-metric-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-metric-space) [Concept guides](/en/derivations/?q=mko-metric-space) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%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 ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-operations) [Concept guides](/en/derivations/?q=mko-set-operations) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Finite discrete probability distribution](/en/mko/mko-finite-discrete-probability-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-discrete-probability-distribution) [Concept guides](/en/derivations/?q=mko-finite-discrete-probability-distribution) - Stage 1 · required ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable) - Stage 1 · required ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 2 · optional ## [Complex modulus and conjugation](/en/mko/mko-complex-modulus-conjugate/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares. [Related questions](/en/explore/?mko=mko-complex-modulus-conjugate) [Concept guides](/en/derivations/?q=mko-complex-modulus-conjugate) - Stage 3 · optional ## [Linear combinations and span](/en/mko/mko-linear-combination-span/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination. [Related questions](/en/explore/?mko=mko-linear-combination-span) [Concept guides](/en/derivations/?q=mko-linear-combination-span) - Stage 4 · optional ## [Subgroups and group homomorphisms](/en/mko/mko-subgroup-homomorphism/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table. [Related questions](/en/explore/?mko=mko-subgroup-homomorphism) [Concept guides](/en/derivations/?q=mko-subgroup-homomorphism) - Stage 5 · optional ## [Convergence in a metric space](/en/mko/mko-metric-convergence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) A sequence in a metric space converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x. [Related questions](/en/explore/?mko=mko-metric-convergence) [Concept guides](/en/derivations/?q=mko-metric-convergence) - Stage 6 · optional ## [Sigma-algebras and measure spaces](/en/mko/mko-sigma-algebra-measure-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) A sigma-algebra contains the empty set and is closed under complements and countable unions; a measure is nonnegative, zero on the empty set and countably additive. [Related questions](/en/explore/?mko=mko-sigma-algebra-measure-space) [Concept guides](/en/derivations/?q=mko-sigma-algebra-measure-space) - Stage 7 · optional ## [Discrete-time stochastic process](/en/mko/mko-discrete-stochastic-process/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law. [Related questions](/en/explore/?mko=mko-discrete-stochastic-process) [Concept guides](/en/derivations/?q=mko-discrete-stochastic-process) - Stage 8 · optional ## [Multivariable real-valued function](/en/mko/mko-multivariable-real-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261023%2F) A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data. [Related questions](/en/explore/?mko=mko-multivariable-real-function) [Concept guides](/en/derivations/?q=mko-multivariable-real-function) Path source & statuspath-pulse-foundations-20261023 · candidate [Original paths JSON](/data/architecture/learning-paths.json)