{"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-20261109","title":{"zh-Hant":"Wave6第三層：Stabilizer、Bolzano-Weierstrass、緊緻子集、全期望、二項式、Möbius與CFG","en":"Wave6 Third Layer: Stabilizers, Bolzano-Weierstrass, Compact Subsets, Total Expectation, Binomial Theorem, Mobius Function and CFGs"},"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-action","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-subgroup-homomorphism","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-subsequence","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-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-supremum-and-infimum","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-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-subspace-topology","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-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-event","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-permutation-combination","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-mathematical-induction","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-polynomial","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-unique-prime-factorization","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-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-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-finite-group-stabilizer","position":2,"requirement":"optional","rationale":{"zh-Hant":"指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。","en":"The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup."}},{"id":"mko-bolzano-weierstrass-theorem-real","position":3,"requirement":"optional","rationale":{"zh-Hant":"每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。","en":"Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map."}},{"id":"mko-compact-subset","position":4,"requirement":"optional","rationale":{"zh-Hant":"K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。","en":"A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover."}},{"id":"mko-law-of-total-expectation","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一有限機率空間上，完整partition各cell的conditional mean按cell probability加權後等於E[X]。","en":"On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]."}},{"id":"mko-binomial-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。","en":"With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients."}},{"id":"mko-mobius-function","position":7,"requirement":"optional","rationale":{"zh-Hant":"Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。","en":"On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign."}},{"id":"mko-context-free-grammar","position":8,"requirement":"optional","rationale":{"zh-Hant":"CFG由有限且不交的nonterminals、terminals、start與單一nonterminal左側的finite productions生成語言。","en":"A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side."}}]},"title":"Wave6第三層：Stabilizer、Bolzano-Weierstrass、緊緻子集、全期望、二項式、Möbius與CFG"}