{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"paths","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{},"path":{"id":"path-pulse-foundations-20261023","title":{"zh-Hant":"現代結構第二批：複數不變量、生成、群同態、收斂、測度、過程與多變數函數","en":"Second Modern Structures: Complex Invariants, Spans, Group Homomorphisms, Convergence, Measure, Processes and Multivariable Functions"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-complex-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-complex-modulus-conjugate","position":2,"requirement":"optional","rationale":{"zh-Hant":"複數共軛保留實部並反轉虛部；模平方為實部與虛部平方和。","en":"Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares."}},{"id":"mko-linear-combination-span","position":3,"requirement":"optional","rationale":{"zh-Hant":"線性組合是有限個向量的純量加權和；生成空間收集全部有限線性組合。","en":"A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination."}},{"id":"mko-subgroup-homomorphism","position":4,"requirement":"optional","rationale":{"zh-Hant":"子群在繼承運算下自身成群；群同態保持乘法，kernel與image由完整映射表導出。","en":"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."}},{"id":"mko-metric-convergence","position":5,"requirement":"optional","rationale":{"zh-Hant":"度量空間序列收斂到x，表示每個正epsilon終有所有n≥N落入x的epsilon球。","en":"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."}},{"id":"mko-sigma-algebra-measure-space","position":6,"requirement":"optional","rationale":{"zh-Hant":"σ-代數含空集並對補集與可數聯集封閉；測度非負、空集為零且可數可加。","en":"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."}},{"id":"mko-discrete-stochastic-process","position":7,"requirement":"optional","rationale":{"zh-Hant":"離散隨機過程是共同機率空間上的離散時間隨機變數族；邊際分布不決定聯合律。","en":"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."}},{"id":"mko-multivariable-real-function","position":8,"requirement":"optional","rationale":{"zh-Hant":"多變數實函數把明示實向量域中的輸入映到實數；座標角色與定義域是函數資料。","en":"A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data."}}]},"title":"現代結構第二批：複數不變量、生成、群同態、收斂、測度、過程與多變數函數"}