{"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-20261101","title":{"zh-Hant":"Wave5根節點：子空間、拓撲覆蓋、幾乎必然收斂、匹配、φ函數與有限自動機","en":"Wave5 Roots: Subspaces, Topological Covers, Almost-Sure Convergence, Matching, Totient and Finite Automata"},"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-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-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-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-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-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-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-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-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-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-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-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-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-prime-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-counting-principle","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-computational-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-discrete-dynamical-system","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-vector-subspace","position":2,"requirement":"optional","rationale":{"zh-Hant":"實向量空間的非空子集若含零向量並對向量加法與每個實純量乘法封閉，則為向量子空間。","en":"A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar."}},{"id":"mko-closure-and-interior","position":3,"requirement":"optional","rationale":{"zh-Hant":"集合閉包是所有閉超集的交，內部是所有開子集的聯；兩者相對於指定拓撲。","en":"The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology."}},{"id":"mko-open-cover-and-subcover","position":4,"requirement":"optional","rationale":{"zh-Hant":"開覆蓋由指定拓撲的開集組成且聯集包含目標集合；子覆蓋須取自原覆蓋並仍覆蓋同一目標。","en":"An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target."}},{"id":"mko-almost-sure-convergence","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一機率空間上的隨機變數序列若在零機率例外集外逐點收斂，則幾乎必然收斂。","en":"Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero."}},{"id":"mko-graph-matching","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限圖的matching是彼此不共享端點的邊子集；空matching合法。","en":"A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed."}},{"id":"mko-euler-totient-function","position":7,"requirement":"optional","rationale":{"zh-Hant":"Euler φ(n)計數一個完整剩餘代表系中與n互質的代表；約定φ(1)=1。","en":"Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1."}},{"id":"mko-finite-automaton","position":8,"requirement":"optional","rationale":{"zh-Hant":"DFA由有限狀態集、有限字母表、總轉移函數、唯一初始狀態與接受狀態子集組成。","en":"A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states."}}]},"title":"Wave5根節點：子空間、拓撲覆蓋、幾乎必然收斂、匹配、φ函數與有限自動機"}