{"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-20261103","title":{"zh-Hant":"Wave5第三層：矩陣秩、函數列逐點收斂、級數、統計模型、連通、最大匹配與模反元素","en":"Wave5 Third Layer: Matrix Rank, Pointwise Function Limits, Series, Statistical Models, Connectedness, Maximum Matching and Modular Inverses"},"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-matrix","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-combination-span","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-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-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-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-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-arithmetic-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-finite-joint-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-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-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-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-open-closed-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-graph-matching","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-optimization-problem","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-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-equation","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":2,"requirement":"optional","rationale":{"zh-Hant":"矩陣秩是row/column space維度，可由精確rank-preserving消去的pivot數計算。","en":"Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination."}},{"id":"mko-pointwise-convergence-of-functions","position":3,"requirement":"optional","rationale":{"zh-Hant":"同一domain上的函數列逐點收斂，若每個固定x的值序列收斂至f(x)，且N可依x與epsilon改變。","en":"A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon."}},{"id":"mko-infinite-series","position":4,"requirement":"optional","rationale":{"zh-Hant":"無窮級數以partial-sum sequence的極限定義；項趨零必要但不充分。","en":"An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence."}},{"id":"mko-parametric-statistical-model","position":5,"requirement":"optional","rationale":{"zh-Hant":"參數統計模型為每個theta指定同一有限可測sample space上的完整機率law P_theta。","en":"A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space."}},{"id":"mko-connected-topological-space","position":6,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間連通，若不存在由兩個不交非空開集組成的separation；等價於無非平凡clopen subset。","en":"A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset."}},{"id":"mko-maximum-matching","position":7,"requirement":"optional","rationale":{"zh-Hant":"最大匹配是在完整有限圖全部合法matching中邊數最大者；可有多個且不必perfect。","en":"A maximum matching has the greatest edge count among all valid matchings of the complete finite graph; it may be nonunique and need not be perfect."}},{"id":"mko-modular-inverse","position":8,"requirement":"optional","rationale":{"zh-Hant":"n≥2時，a模n有乘法反元素恰當gcd(a,n)=1；反元素在模n意義唯一。","en":"For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n."}}]},"title":"Wave5第三層：矩陣秩、函數列逐點收斂、級數、統計模型、連通、最大匹配與模反元素"}