{"schema_version":"ocme-site-results-v1","locale":"en","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-20261102","title":{"zh-Hant":"Wave5第二層：維度、核像、拓撲邊界、緊緻、Bayes與正規語言","en":"Wave5 Second Layer: Dimension, Kernels, Images, Topological Boundary, Compactness, Bayes and Regular Languages"},"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-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-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-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-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-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-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-closure-and-interior","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-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-cover-and-subcover","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-conditional-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-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-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-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-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-decision-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-vector-space-dimension","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限維向量空間的維度是任一有限基底的元素數；基底不同但基數不變。","en":"The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality."}},{"id":"mko-linear-map-kernel","position":3,"requirement":"optional","rationale":{"zh-Hant":"線性映射T的核是全部滿足T(x)=0的來源向量，並形成來源子空間。","en":"The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source."}},{"id":"mko-linear-map-image","position":4,"requirement":"optional","rationale":{"zh-Hant":"線性映射的像是所有輸出T(x)的集合；矩陣映射的像等於column span。","en":"The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span."}},{"id":"mko-topological-boundary","position":5,"requirement":"optional","rationale":{"zh-Hant":"集合A的拓撲邊界為closure(A)減去interior(A)，亦等於closure(A)交closure(complement A)。","en":"The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement."}},{"id":"mko-compact-topological-space","position":6,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間緊緻，若每個開覆蓋都有有限子覆蓋；每個有限拓撲空間皆緊緻。","en":"A space is compact when every open cover has a finite subcover; every finite topological space is compact."}},{"id":"mko-bayes-theorem","position":7,"requirement":"optional","rationale":{"zh-Hant":"正分母條件下，Bayes定理以共同intersection與priors/likelihoods精確反轉條件機率。","en":"With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods."}},{"id":"mko-regular-language","position":8,"requirement":"optional","rationale":{"zh-Hant":"正規語言是存在一個完整DFA可辨識其全部字串的語言。","en":"A language is regular when a complete DFA recognizes all and exactly its words."}}]},"title":"Wave5 Second Layer: Dimension, Kernels, Images, Topological Boundary, Compactness, Bayes and Regular Languages"}