{"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-20261020","title":{"zh-Hant":"應用數學基礎第一批：動力、微分方程、數值、最佳化、物理、生物與金融","en":"First Applied Foundations: Dynamics, ODEs, Numerical Error, Optimization, Physics, Biology, and Finance"},"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-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-natural-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-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-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-absolute-value","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-ratio-proportion","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-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-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-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-discrete-dynamical-system","position":2,"requirement":"optional","rationale":{"zh-Hant":"離散時間動力系統以狀態空間、自映射與自然數時間描述逐步演化；有限軌道核對不自動給出一般穩定性。","en":"A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability."}},{"id":"mko-ordinary-differential-equation","position":3,"requirement":"optional","rationale":{"zh-Hant":"常微分方程關聯未知單變數函數及其有限階導數；候選函數的殘差核對不等於存在唯一性定理。","en":"An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems."}},{"id":"mko-numerical-approximation-error","position":4,"requirement":"optional","rationale":{"zh-Hant":"絕對誤差是近似值與參照值之差的絕對值；相對誤差另要求參照值非零並除以其絕對值。","en":"Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value."}},{"id":"mko-optimization-problem","position":5,"requirement":"optional","rationale":{"zh-Hant":"最佳化問題在明示可行集合上尋找使目標函數最小或最大的點；最優值與最優解集合需分開。","en":"An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects."}},{"id":"mko-dimensional-analysis","position":6,"requirement":"optional","rationale":{"zh-Hant":"量綱分析用基本量綱指數向量追蹤乘、除與冪，並核對方程兩側量綱是否一致。","en":"Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides."}},{"id":"mko-discrete-exponential-growth","position":7,"requirement":"optional","rationale":{"zh-Hant":"固定時間步與固定非負因子 r 的離散模型滿足 x_(n+1)=r x_n，因此 x_n=x_0 r^n。","en":"With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n."}},{"id":"mko-compound-interest","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定每期利率 i 且無額外現金流時，餘額滿足 B_(n+1)=(1+i)B_n，故 B_n=B_0(1+i)^n。","en":"With a fixed rate i per period and no additional cash flows, balances satisfy B_(n+1)=(1+i)B_n and hence B_n=B_0(1+i)^n."}}]},"title":"First Applied Foundations: Dynamics, ODEs, Numerical Error, Optimization, Physics, Biology, and Finance"}