{"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-20261021","title":{"zh-Hant":"應用數學基礎第二批：表徵、診斷、週期、初值、二分、線性規劃與受限成長","en":"Second Applied Foundations: Representation, Assessment Records, Periodicity, Initial Values, Bisection, Linear Programming and Capacity-Limited Growth"},"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-variable-expression","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-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-proposition","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-counterexample","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-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-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-domain-codomain-range","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-numerical-approximation-error","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-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-optimization-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-linear-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-system-linear-equations","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-discrete-exponential-growth","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-mathematical-representation","position":2,"requirement":"optional","rationale":{"zh-Hant":"數學表徵把同一對象的文字規則、符號式、有限表格、序對與座標資料分開記錄，並以明示語義橋核對一致性。","en":"Mathematical representation separately records verbal rules, symbolic expressions, finite tables, ordered pairs and coordinate data for the same object, using explicit semantic bridges to check agreement."}},{"id":"mko-diagnostic-mathematical-assessment","position":3,"requirement":"optional","rationale":{"zh-Hant":"診斷性數學評量記錄把題目、期望、作答與明示判定規則綁在一起，只產生可追溯的規則符合狀態。","en":"A diagnostic mathematical assessment record binds an item, its expected result, a response and an explicit judgment rule, producing only a traceable rule-agreement status."}},{"id":"mko-orbit-fixed-periodic","position":4,"requirement":"optional","rationale":{"zh-Hant":"反覆迭代自映射形成軌道；固定點週期為1，週期點的最小正週期須和任意返回時間分開。","en":"Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time."}},{"id":"mko-initial-value-problem","position":5,"requirement":"optional","rationale":{"zh-Hant":"初值問題在微分方程外指定某點的函數值或導數值；候選核對與存在唯一性是不同責任。","en":"An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness."}},{"id":"mko-bisection-method","position":6,"requirement":"optional","rationale":{"zh-Hant":"二分法在已提供的端點與中點符號資料上取中點並保留符號相反或零值的半區，有限步寬度減半。","en":"Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width."}},{"id":"mko-linear-programming","position":7,"requirement":"optional","rationale":{"zh-Hant":"線性規劃在線性等式或不等式限制的可行域上最佳化線性目標函數；本輪題庫只驗完整有限候選網格。","en":"Linear programming optimizes a linear objective over a feasible domain defined by linear equalities or inequalities; this exercise slice checks only a complete explicit finite candidate grid."}},{"id":"mko-discrete-logistic-growth","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定容量K與0≤r≤1時，本條目以x_(n+1)=x_n+r x_n(1-x_n/K)描述每步受限增量。","en":"For fixed capacity K and 0≤r≤1, this entry uses x_(n+1)=x_n+r*x_n*(1−x_n/K) to describe each capacity-limited discrete increment."}}]},"title":"Second Applied Foundations: Representation, Assessment Records, Periodicity, Initial Values, Bisection, Linear Programming and Capacity-Limited Growth"}