{"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-20261110","title":{"zh-Hant":"Wave6第四層：Orbit-Stabilizer、MVT、緊緻像、全變異數、遞迴、Möbius反演與PDA","en":"Wave6 Fourth Layer: Orbit-Stabilizer, MVT, Compact Images, Total Variance, Recurrences, Mobius Inversion and PDAs"},"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-finite-group-orbit","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-stabilizer","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-cardinality-comparison","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-derivative","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-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-rate-of-change","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-order-relation","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-compact-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-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-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-law-of-total-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-conditional-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-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-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-mobius-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-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-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-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-finite-automaton","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-context-free-grammar","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-orbit-stabilizer-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。","en":"For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order."}},{"id":"mko-mean-value-theorem","position":3,"requirement":"optional","rationale":{"zh-Hant":"若a<b，f在[a,b]連續且在(a,b)可微，則某個內點c的導數等於端點割線斜率。","en":"If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope."}},{"id":"mko-continuous-image-of-compact","position":4,"requirement":"optional","rationale":{"zh-Hant":"若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。","en":"If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y."}},{"id":"mko-law-of-total-variance","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一有限law與完整partition生成的G下，總變異數等於cell內變異數期望加cell均值變異數。","en":"For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means."}},{"id":"mko-recurrence-relation","position":6,"requirement":"optional","rationale":{"zh-Hant":"order-r recurrence以明示initial values與total single-valued update rule逐項決定sequence。","en":"An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms."}},{"id":"mko-mobius-inversion-finite-divisors","position":7,"requirement":"optional","rationale":{"zh-Hant":"若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。","en":"If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n)."}},{"id":"mko-pushdown-automaton","position":8,"requirement":"optional","rationale":{"zh-Hant":"nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。","en":"A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state."}}]},"title":"Wave6第四層：Orbit-Stabilizer、MVT、緊緻像、全變異數、遞迴、Möbius反演與PDA"}