{"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":"12"},"kind":"guides","total":299,"pages":13,"page":12,"entries":[{"id":"derivation-mko-estimator-mean-squared-error-01","title":{"zh-Hant":"不用分解公式的直接MSE計算","en":"Direct MSE without the decomposition formula"},"summary":{"zh-Hant":"逐theta直接加權loss，保留bias-variance為後續。","en":"Weight losses directly at every theta, leaving decomposition for later."},"mko_ids":["mko-estimator-mean-squared-error"]},{"id":"derivation-mko-linear-program-standard-form-01","title":{"zh-Hant":"slack 的雙向點對應","en":"The two-way point correspondence for slack variables"},"summary":{"zh-Hant":"把固定 maximize/≤/nonnegative LP 轉為 equality slack form，並核雙向可行點與 objective。","en":"Convert a fixed maximize/≤/nonnegative LP to equality slack form and verify both feasible-point directions and objective."},"mko_ids":["mko-linear-program-standard-form"]},{"id":"derivation-mko-vertex-cover-01","title":{"zh-Hant":"星形圖的cover與刪點證書","en":"Vertex covers and deletion certificates on a star"},"summary":{"zh-Hant":"逐邊核cover並區分minimal與minimum。","en":"Audit covers edgewise and distinguish minimal from minimum."},"mko_ids":["mko-vertex-cover"]},{"id":"derivation-mko-first-isomorphism-theorem-finite-groups-01","title":{"zh-Hant":"非滿射同態的正確image同構","en":"The correct image isomorphism for a nonsurjection"},"summary":{"zh-Hant":"f:C₄→C₄,f(a)=2a mod4給C₄/{0,2}≅{0,2}，不是整個codomain C₄。","en":"For f:C4 to C4,f(a)=2a mod 4, C4/{0,2} is isomorphic to {0,2}, not the full codomain C4."},"mko_ids":["mko-first-isomorphism-theorem-finite-groups"]},{"id":"derivation-mko-finite-group-center-01","title":{"zh-Hant":"全部交換rows認證群中心","en":"Certifying the center from all commutation rows"},"summary":{"zh-Hant":"Z(G)={e}，直接驗subgroup及normal；G非abelian，r雖與自身交換卻不central。","en":"Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself."},"mko_ids":["mko-finite-group-center"]},{"id":"derivation-mko-bias-variance-decomposition-01","title":{"zh-Hant":"展開cross term的bias-variance推導","en":"Bias-variance derivation with the cross term visible"},"summary":{"zh-Hant":"在同一law逐符號證明分解，並核兩個theta。","en":"Prove the decomposition symbol by symbol in one law and check both theta values."},"mko_ids":["mko-bias-variance-decomposition"]},{"id":"derivation-mko-linear-program-duality-01","title":{"zh-Hant":"弱對偶鏈與零間隙最優證書","en":"The weak-duality chain and a zero-gap optimality certificate"},"summary":{"zh-Hant":"完整構造對偶，核 x=(1,3)、y=(1,1) 可行並由零間隙證雙最優。","en":"Construct the full dual, check feasibility of x=(1,3), y=(1,1), and certify both optimal by zero gap."},"mko_ids":["mko-linear-program-duality"]},{"id":"derivation-mko-max-flow-min-cut-theorem-01","title":{"zh-Hant":"完整四個 cut 與等值雙證書","en":"All four cuts and an equal-value two-sided certificate"},"summary":{"zh-Hant":"枚舉四點網路全部 s-t cuts，給值 5 的可行流與同值 cut。","en":"Enumerate every s-t cut of a four-vertex network and give a feasible flow of value 5 together with a cut of capacity 5."},"mko_ids":["mko-max-flow-min-cut-theorem"]},{"id":"derivation-mko-minimum-vertex-cover-01","title":{"zh-Hant":"P3全部八子集求minimum cover","en":"All eight subsets for minimum cover on P3"},"summary":{"zh-Hant":"完整分類所有covers與全部minimizers。","en":"Classify every cover and all minimizers."},"mko_ids":["mko-minimum-vertex-cover"]},{"id":"derivation-mko-many-one-reduction-01","title":{"zh-Hant":"完整有限reduction與方向保持","en":"A complete finite reduction with preserved direction"},"summary":{"zh-Hant":"逐x核iff、組合及decider transfer。","en":"Check every x equivalence, composition and decider transfer."},"mko_ids":["mko-many-one-reduction"]},{"id":"derivation-mko-contraction-mapping-theorem-01","title":{"zh-Hant":"有限度量的完整收縮證書","en":"A complete contraction certificate on a finite metric"},"summary":{"zh-Hant":"在三點離散度量與常值 self-map 上核完備、唯一不動點及誤差界。","en":"Audit completeness, a unique fixed point and the error bound for a constant self-map on a three-point discrete metric."},"mko_ids":["mko-contraction-mapping-theorem"]},{"id":"derivation-mko-intermediate-value-theorem-01","title":{"zh-Hant":"多項式 bracket 與精確中間值","en":"A polynomial bracket and an exact intermediate value"},"summary":{"zh-Hant":"對 f(x)=x^2-1 在 [0,2] 找值 0，並保留端點及非唯一邊界。","en":"Find value zero for f(x)=x^2-1 on [0,2], retaining endpoint and nonuniqueness boundaries."},"mko_ids":["mko-intermediate-value-theorem"]},{"id":"derivation-mko-taylor-remainder-theorem-01","title":{"zh-Hant":"三次多項式的精確 Lagrange 餘項","en":"An exact Lagrange remainder for a cubic"},"summary":{"zh-Hant":"以 f(x)=x^3、a=1、n=2、x=2 核精確餘項及有效導數界。","en":"For f(x)=x^3, a=1, n=2, x=2, audit the exact remainder and a qualified derivative bound."},"mko_ids":["mko-taylor-remainder-theorem"]},{"id":"derivation-mko-finite-maximum-likelihood-estimator-01","title":{"zh-Hant":"含正質量 tie 與全零似然欄的 MLE","en":"MLE with a positive-mass tie and an all-zero likelihood column"},"summary":{"zh-Hant":"完整算argmax及固定tie rule的total estimator。","en":"Compute every argmax and a total estimator using a fixed tie rule."},"mko_ids":["mko-finite-maximum-likelihood-estimator"]},{"id":"derivation-mko-finite-sufficient-statistic-01","title":{"zh-Hant":"逐fiber條件law的充分性證書","en":"A sufficiency certificate from every fibre conditional law"},"summary":{"zh-Hant":"完整核正質量fiber跨theta不變，保留null未定義。","en":"Audit invariance across theta on every positive-mass fibre, retaining undefined null conditions."},"mko_ids":["mko-finite-sufficient-statistic"]},{"id":"derivation-mko-topological-ordering-dag-01","title":{"zh-Hant":"三頂點DAG的全部排列與Kahn trace","en":"All permutations and a Kahn trace for a three-vertex DAG"},"summary":{"zh-Hant":"完整列valid orders並核source-removal invariant。","en":"List every valid order and audit source-removal invariants."},"mko_ids":["mko-topological-ordering-dag"]},{"id":"derivation-mko-multiplicative-order-mod-n-01","title":{"zh-Hant":"第一個回到1的冪","en":"The first power returning to one"},"summary":{"zh-Hant":"ord₈(−5)=ord₈(3)=2，φ(8)=4且2整除4，但order不等φ。","en":"ord_8(-5)=ord_8(3)=2, with phi(8)=4 and 2 dividing 4 but not equal to it."},"mko_ids":["mko-multiplicative-order-mod-n"]},{"id":"derivation-mko-series-comparison-test-01","title":{"zh-Hant":"最終比較的方向與有限前綴","en":"The direction of eventual comparison and finite prefixes"},"summary":{"zh-Hant":"證一個有任意有限前綴的級數由幾何尾上比較收斂，並示下比較發散。","en":"Prove convergence using geometric upper comparison despite a finite prefix, and show divergent lower comparison."},"mko_ids":["mko-series-comparison-test"]},{"id":"derivation-mko-ratio-test-series-01","title":{"zh-Hant":"比值極限、幾何支配與 L=1","en":"Ratio limits, geometric domination and L=1"},"summary":{"zh-Hant":"以完整有理比值證 L<1 收斂及 L>1 發散，並核 L=1 無結論。","en":"Use complete rational ratios for L<1 convergence and L>1 divergence, then audit indecision at L=1."},"mko_ids":["mko-ratio-test-series"]},{"id":"derivation-mko-uniform-limit-continuity-theorem-01","title":{"zh-Hant":"固定一個近似函數的 epsilon/3 傳遞","en":"An epsilon/3 transfer through one fixed approximant"},"summary":{"zh-Hant":"對 [0,1] 上的仿射擾動證一致收斂，並逐步傳遞連續性。","en":"Prove uniform convergence of affine perturbations on [0,1] and transfer continuity step by step."},"mko_ids":["mko-uniform-limit-continuity-theorem"]},{"id":"derivation-mko-finite-product-topology-01","title":{"zh-Hant":"四點乘積的全部開集","en":"Every open set in a four-point product"},"summary":{"zh-Hant":"由兩個 Sierpinski 因子生成全部乘積開集並核投影。","en":"Generate every product open from two Sierpinski factors and audit projections."},"mko_ids":["mko-finite-product-topology"]},{"id":"derivation-mko-quotient-topology-01","title":{"zh-Hant":"完整逆像條件決定商拓撲","en":"The full preimage condition determines the quotient topology"},"summary":{"zh-Hant":"由三點到兩點的滿射計算商拓撲與開像失敗。","en":"Compute a quotient topology for a three-to-two-point surjection and show an open-image failure."},"mko_ids":["mko-quotient-topology"]},{"id":"derivation-mko-path-connected-space-01","title":{"zh-Hant":"凸盒中的全稱路徑與拼接","en":"Universal paths and concatenation in a convex box"},"summary":{"zh-Hant":"證 [0,1]^2 路徑連通，並核兩段路徑的接點。","en":"Prove [0,1]^2 path connected and audit a two-piece concatenation."},"mko_ids":["mko-path-connected-space"]},{"id":"derivation-mko-topological-connected-component-01","title":{"zh-Hant":"完整子集表找包含極大的連通分量","en":"Finding inclusion-maximal components from the complete subset table"},"summary":{"zh-Hant":"求 X={a,b,c}、tau={空集,{a},{b,c},X} 的全部連通分量。","en":"Find every component of X={a,b,c}, tau={empty,{a},{b,c},X}."},"mko_ids":["mko-topological-connected-component"]}],"title":"概念推導導讀"}