{"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-20261104","title":{"zh-Hant":"Wave5第四層：rank-nullity、一致收斂、絕對收斂、估計量、CRT、生成樹與Lp收斂","en":"Wave5 Fourth Layer: Rank-Nullity, Uniform and Absolute Convergence, Estimators, CRT, Spanning Trees and Lp Limits"},"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-linear-map-kernel","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-linear-map-image","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-matrix-rank","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-space-dimension","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-pointwise-convergence-of-functions","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-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-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-infinite-series","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-limit","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-parametric-statistical-model","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-sample-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-integer","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-modular-congruence","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-gcd-coprime","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-tree","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-graph","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-connected-component","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-expectation","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-sigma-algebra-measure-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-rank-nullity-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限維線性映射T:V→W滿足dim V=dim ker T+rank T。","en":"A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T)."}},{"id":"mko-uniform-convergence-of-functions","position":3,"requirement":"optional","rationale":{"zh-Hant":"一致收斂要求同一N(epsilon)同時控制domain全部x。","en":"Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously."}},{"id":"mko-absolute-convergence-of-series","position":4,"requirement":"optional","rationale":{"zh-Hant":"sum |a_n|收斂時sum a_n絕對收斂；絕對收斂推出收斂，反向不一般成立。","en":"A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general."}},{"id":"mko-statistical-estimator","position":5,"requirement":"optional","rationale":{"zh-Hant":"估計量是固定且不依未知theta的可測total statistic；觀察後數值是estimate。","en":"An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate."}},{"id":"mko-chinese-remainder-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"正兩兩互質模數的同餘系統在模乘積意義下存在唯一解。","en":"For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product."}},{"id":"mko-spanning-tree","position":7,"requirement":"optional","rationale":{"zh-Hant":"生成樹是包含父圖全部頂點的連通無圈子圖。","en":"A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph."}},{"id":"mko-lp-convergence","position":8,"requirement":"optional","rationale":{"zh-Hant":"共同機率空間上固定整數p≥1時，E|X_n−X|^p→0稱Lp收斂。","en":"On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero."}}]},"title":"Wave5第四層：rank-nullity、一致收斂、絕對收斂、估計量、CRT、生成樹與Lp收斂"}