{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","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":{"page":"13"},"kind":"guides","total":299,"pages":13,"page":13,"entries":[{"id":"derivation-mko-central-limit-theorem-finite-variance-iid-01","title":{"zh-Hant":"有限variance IID的CLT資格與格點audit","en":"CLT qualification and a finite lattice audit"},"summary":{"zh-Hant":"清楚區分finite-n moments與named CLT應用。","en":"Separate finite-n moments from the named CLT application."},"mko_ids":["mko-central-limit-theorem-finite-variance-iid"]},{"id":"derivation-mko-strong-law-large-numbers-finite-valued-iid-01","title":{"zh-Hant":"有限值IID的強律資格與固定n界線","en":"Finite-valued IID strong-law qualification and fixed-n boundaries"},"summary":{"zh-Hant":"對齊named SLLN前提，並明示prefix不是a.s.證明。","en":"Match the named SLLN hypotheses while separating prefix audits from a.s. proof."},"mko_ids":["mko-strong-law-large-numbers-finite-valued-iid"]},{"id":"derivation-mko-finite-confidence-interval-coverage-01","title":{"zh-Hant":"逐參數coverage、平均與selection","en":"Pointwise coverage, averaging and selection"},"summary":{"zh-Hant":"完整計算固定區間程序的pointwise及worst-case coverage。","en":"Compute pointwise and worst-case coverage of a fixed interval procedure."},"mko_ids":["mko-finite-confidence-interval-coverage"]},{"id":"derivation-mko-wilson-theorem-01","title":{"zh-Hant":"完整逆元配對與Wilson雙向證明","en":"Complete inverse pairing and both Wilson directions"},"summary":{"zh-Hant":"n>1時(n−1)!≡−1 modn恰當n prime；模7完整逆元表核6!≡6。","en":"For n>1, (n-1)!=-1 mod n exactly when n is prime; the full modulo-7 inverse table checks 6!=6 mod 7."},"mko_ids":["mko-wilson-theorem"]},{"id":"derivation-mko-p-adic-valuation-integers-01","title":{"zh-Hant":"帶符號整數的3進因子抽取","en":"Extracting 3-adic factors of signed integers"},"summary":{"zh-Hant":"v₃(−108)=3、v₃(18)=2、v₃(−1944)=5、v₃(−90)=2；零用infinity。","en":"v3(-108)=3,v3(18)=2,v3(-1944)=5,v3(-90)=2; zero uses infinity."},"mko_ids":["mko-p-adic-valuation-integers"]},{"id":"derivation-mko-regular-expression-01","title":{"zh-Hant":"含epsilon的star閉包與完整長度切片","en":"Star closure with epsilon and a complete length slice"},"summary":{"zh-Hant":"typed AST遞迴語義、完整bounded枚舉與membership責任。","en":"Give typed AST semantics, complete bounded enumeration and membership obligations."},"mko_ids":["mko-regular-expression"]},{"id":"derivation-mko-pumping-lemma-regular-languages-01","title":{"zh-Hant":"DFA迴圈量詞與a^n b^n反證","en":"DFA-loop quantifiers and the a^n b^n contradiction"},"summary":{"zh-Hant":"可見量詞次序、全pump count迴圈證明及全部合法splits。","en":"Expose quantifier order, the all-pump-count loop argument and every legal split."},"mko_ids":["mko-pumping-lemma-regular-languages"]},{"id":"derivation-mko-newton-method-01","title":{"zh-Hant":"合法 Newton 更新與有限 trace","en":"Qualified Newton updates and a finite trace"},"summary":{"zh-Hant":"對 x^2-2 從 1 做兩步精確 Newton，並核 root-first 與循環邊界。","en":"Take two exact Newton steps for x^2-2 from one, and audit root-first stopping and a cycle boundary."},"mko_ids":["mko-newton-method"]},{"id":"derivation-mko-trapezoidal-quadrature-01","title":{"zh-Hant":"非均勻梯形值與帶符號誤差","en":"Nonuniform trapezoids and signed error"},"summary":{"zh-Hant":"對 f(x)=x^2、nodes (0,1/2,2) 計算 T、I-T 與絕對誤差。","en":"For f(x)=x^2 and nodes (0,1/2,2), compute T, I-T and absolute error."},"mko_ids":["mko-trapezoidal-quadrature"]},{"id":"derivation-mko-gradient-descent-01","title":{"zh-Hant":"正對角二次函數的安全下降步","en":"Safe descent steps for a positive diagonal quadratic"},"summary":{"zh-Hant":"以 alpha=1/4 對指定二次函數做兩步並證每步下降條件。","en":"Take two steps with alpha=1/4 for a specified quadratic and prove the per-step descent condition."},"mko_ids":["mko-gradient-descent"]},{"id":"derivation-mko-lagrange-multiplier-condition-01","title":{"zh-Hant":"正則乘數與完整平方距離解","en":"Regular multipliers and a complete squared-distance solution"},"summary":{"zh-Hant":"minimize ‖x-(1,2)‖^2 subject to x_1+x_2=0，求點、乘數並獨立證最小。","en":"Minimize the squared distance to (1,2) subject to x_1+x_2=0; find the point and multiplier and independently prove the minimum."},"mko_ids":["mko-lagrange-multiplier-condition"]}],"title":"概念推導導讀"}