{"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-20261022","title":{"zh-Hant":"現代結構第一批：振子、折現、複數、向量空間、有限群、度量與分布","en":"First Modern Structures: Oscillators, Discounting, Complex Numbers, Vector Spaces, Finite Groups, Metrics and Distributions"},"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-ordinary-differential-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-derivative-rules","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-power-root","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-compound-interest","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-fraction","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-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-coordinate-plane","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-vector","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-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-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-proof","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-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-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-harmonic-oscillator-model","position":2,"requirement":"optional","rationale":{"zh-Hant":"無阻尼線性簡諧振子以m x''+k x=0及m,k>0定義；本輪不宣稱三角函數閉式解。","en":"The undamped linear harmonic oscillator is defined by m*x″+k*x=0 with m,k>0; this slice asserts no trigonometric closed-form solution."}},{"id":"mko-present-value-discounting","position":3,"requirement":"optional","rationale":{"zh-Hant":"固定正累積因子與期數時，未來金額除以累積因子冪得到現值。","en":"For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value."}},{"id":"mko-complex-number","position":4,"requirement":"optional","rationale":{"zh-Hant":"複數以實數有序對(a,b)表示a+bi，並以i²=-1定義加減乘法。","en":"A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1."}},{"id":"mko-real-vector-space","position":5,"requirement":"optional","rationale":{"zh-Hant":"實向量空間配備向量加法與實數純量乘法，滿足加法交換群及相容的純量作用公理。","en":"A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms."}},{"id":"mko-finite-group","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限群由有限集合與完整封閉Cayley表構成，並滿足結合律、唯一單位元及每元素逆元。","en":"A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element."}},{"id":"mko-metric-space","position":7,"requirement":"optional","rationale":{"zh-Hant":"度量空間由集合與距離函數組成，距離非負、只在相同點為零、對稱且滿足三角不等式。","en":"A metric space has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality."}},{"id":"mko-finite-discrete-probability-distribution","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限離散分布是在有限結果值上指定非負機率質量，全部質量精確和為一。","en":"A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one."}}]},"title":"現代結構第一批：振子、折現、複數、向量空間、有限群、度量與分布"}