{"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-20261105","title":{"zh-Hant":"Wave5第五層：直和、對偶、Euclidean演算法、Euler trail、同胚、抽樣分布與弱大數律","en":"Wave5 Fifth Layer: Direct Sums, Duals, Euclid, Euler Trails, Homeomorphisms, Sampling Laws and the WLLN"},"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-vector-subspace","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-linear-independence-basis","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-general-linear-map","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-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-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-natural-divisibility","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-algorithm-correctness","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-walk-simple-path","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-topological-continuity","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-bijective","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-inverse-function","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-topological-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-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-statistical-estimator","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-iid-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-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-variance","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-convergence-in-probability","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-direct-sum","position":2,"requirement":"optional","rationale":{"zh-Hant":"二個子空間之和若每個向量有唯一分解便是內直和；要宣稱覆蓋母空間還須另證生成。","en":"The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation."}},{"id":"mko-dual-vector-space","position":3,"requirement":"optional","rationale":{"zh-Hant":"實向量空間的對偶空間由所有實值線性泛函組成；座標covector表示依賴基底。","en":"The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis."}},{"id":"mko-euclidean-algorithm","position":4,"requirement":"optional","rationale":{"zh-Hant":"Euclidean演算法以精確帶餘除法、gcd invariant與嚴格餘數下降求最大公因數。","en":"The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders."}},{"id":"mko-eulerian-trail","position":5,"requirement":"optional","rationale":{"zh-Hant":"Euler trail恰用每條邊一次；非零邊支撐連通且奇度數為0或2時分別有circuit或open trail。","en":"An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail."}},{"id":"mko-homeomorphism","position":6,"requirement":"optional","rationale":{"zh-Hant":"同胚是雙射、連續且逆映射亦連續的映射；一般continuous bijection仍可能不是同胚。","en":"A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism."}},{"id":"mko-sampling-distribution","position":7,"requirement":"optional","rationale":{"zh-Hant":"固定theta時，估計量把樣本分布pushforward成統計量的抽樣分布；全部theta形成一族laws。","en":"At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws."}},{"id":"mko-weak-law-large-numbers","position":8,"requirement":"optional","rationale":{"zh-Hant":"IID且具有有限variance時，Chebyshev界給樣本平均依機率收斂到共同mean。","en":"For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean."}}]},"title":"Wave5第五層：直和、對偶、Euclidean演算法、Euler trail、同胚、抽樣分布與弱大數律"}