{"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":"8"},"kind":"guides","total":299,"pages":13,"page":8,"entries":[{"id":"derivation-mko-graph-coloring-01","title":{"zh-Hant":"完整著色與非最優greedy trace","en":"A complete coloring and a nonoptimal greedy trace"},"summary":{"zh-Hant":"逐邊驗proper coloring並展示上界非自動最優。","en":"Audit a proper coloring edgewise and show an upper bound need not be optimal."},"mko_ids":["mko-graph-coloring"]},{"id":"derivation-mko-vector-subspace-01","title":{"zh-Hant":"以全稱封閉認證過原點直線","en":"Universal closure of a line through zero"},"summary":{"zh-Hant":"U={(x,2x):x∈R}是R²子空間，仿射平移U+(0,1)不是。","en":"U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not."},"mko_ids":["mko-vector-subspace"]},{"id":"derivation-mko-closure-and-interior-01","title":{"zh-Hant":"在鏈狀三點拓撲求閉包與內部","en":"Closure and interior in a three-point chain topology"},"summary":{"zh-Hant":"求 A={b} 的閉包及內部。","en":"Find the closure and interior of A={b}."},"mko_ids":["mko-closure-and-interior"]},{"id":"derivation-mko-open-cover-and-subcover-01","title":{"zh-Hant":"從原覆蓋索引選取子覆蓋","en":"Selecting a subcover by original indices"},"summary":{"zh-Hant":"核對 K={a,b} 的指定開覆蓋與子覆蓋索引。","en":"Audit a supplied open cover of K={a,b} and its subcover indices."},"mko_ids":["mko-open-cover-and-subcover"]},{"id":"derivation-mko-almost-sure-convergence-01","title":{"zh-Hant":"逐原子尾端與零質量例外","en":"Atomwise tails and a null exception"},"summary":{"zh-Hant":"完整列出每個原子的收斂或失敗證書。","en":"Give a convergence or failure certificate for every atom."},"mko_ids":["mko-almost-sure-convergence"]},{"id":"derivation-mko-graph-matching-01","title":{"zh-Hant":"P4中maximal與maximum的分界","en":"Maximal versus maximum in P4"},"summary":{"zh-Hant":"完整核選邊與端點，再分清三種匹配性質。","en":"Audit selected edges and endpoints, then distinguish matching properties."},"mko_ids":["mko-graph-matching"]},{"id":"derivation-mko-euler-totient-function-01","title":{"zh-Hant":"逐代表計算φ(12)","en":"Counting phi(12) representative by representative"},"summary":{"zh-Hant":"φ(12)=4，φ(1)=1，且本例φ(3·4)=φ(3)φ(4)。","en":"phi(12)=4, phi(1)=1, and here phi(3*4)=phi(3)*phi(4)."},"mko_ids":["mko-euler-totient-function"]},{"id":"derivation-mko-finite-automaton-01","title":{"zh-Hant":"完整DFA的1011追蹤","en":"Tracing 1011 through a complete DFA"},"summary":{"zh-Hant":"指定DFA拒絕1011，並接受空字ε。","en":"The specified DFA rejects 1011 and accepts epsilon."},"mko_ids":["mko-finite-automaton"]},{"id":"derivation-mko-vector-space-dimension-01","title":{"zh-Hant":"先證基底再讀取維度","en":"Prove a basis before reading dimension"},"summary":{"zh-Hant":"B=((1,1),(1,−1))是R²有序基底，故dim R²=2。","en":"B=((1,1),(1,-1)) is an ordered basis of R2, so dim R2=2."},"mko_ids":["mko-vector-space-dimension"]},{"id":"derivation-mko-linear-map-kernel-01","title":{"zh-Hant":"完整求解二階矩陣的核","en":"The full kernel of a two-by-two map"},"summary":{"zh-Hant":"T(x,y)=(x+2y,2x+4y)的核為span{(−2,1)}。","en":"The kernel of T(x,y)=(x+2y,2x+4y) is span{(-2,1)}."},"mko_ids":["mko-linear-map-kernel"]},{"id":"derivation-mko-linear-map-image-01","title":{"zh-Hant":"用一致性求完整像空間","en":"The full image from consistency"},"summary":{"zh-Hant":"im T=span{(1,2)}；(3,6)在像內，(0,1)不在像內。","en":"im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not."},"mko_ids":["mko-linear-map-image"]},{"id":"derivation-mko-topological-boundary-01","title":{"zh-Hant":"邊界與補集的精確對照","en":"Exact boundary comparison with the complement"},"summary":{"zh-Hant":"計算鏈狀拓撲中 A={b} 與其補集的邊界。","en":"Compute the boundaries of A={b} and its complement in a chain topology."},"mko_ids":["mko-topological-boundary"]},{"id":"derivation-mko-compact-topological-space-01","title":{"zh-Hant":"有限拓撲的全稱緊緻證書","en":"A universal compactness certificate for a finite topology"},"summary":{"zh-Hant":"證明三點鏈狀拓撲對每個開覆蓋都有有限子覆蓋。","en":"Prove every open cover of the three-point chain topology has a finite subcover."},"mko_ids":["mko-compact-topological-space"]},{"id":"derivation-mko-bayes-theorem-01","title":{"zh-Hant":"四原子檢測模型的Bayes反轉","en":"Bayes reversal in a four-atom test model"},"summary":{"zh-Hant":"由完整joint law算prior、likelihood與posterior。","en":"Compute priors, likelihoods and posteriors from a complete joint law."},"mko_ids":["mko-bayes-theorem"]},{"id":"derivation-mko-regular-language-01","title":{"zh-Hant":"任意字長的parity invariant","en":"A parity invariant for arbitrary word length"},"summary":{"zh-Hant":"Σ={0,1}上含偶數個1的全部word構成正規語言。","en":"All words over {0,1} with an even number of 1s form a regular language."},"mko_ids":["mko-regular-language"]},{"id":"derivation-mko-matrix-rank-01","title":{"zh-Hant":"三階矩陣的完整RREF與上下界","en":"Full RREF and rank bounds"},"summary":{"zh-Hant":"A=[[1,2,3],[2,4,6],[1,1,1]]的row與column rank均為2。","en":"A=[[1,2,3],[2,4,6],[1,1,1]] has row and column rank 2."},"mko_ids":["mko-matrix-rank"]},{"id":"derivation-mko-pointwise-convergence-of-functions-01","title":{"zh-Hant":"有限定義域的逐點尾項證書","en":"Pointwise tail certificates on a finite domain"},"summary":{"zh-Hant":"證 f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) 逐點趨於 f。","en":"Prove pointwise convergence of f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) to f."},"mko_ids":["mko-pointwise-convergence-of-functions"]},{"id":"derivation-mko-infinite-series-01","title":{"zh-Hant":"幾何與望遠鏡級數的完整尾項","en":"Complete tails for geometric and telescoping series"},"summary":{"zh-Hant":"用精確部分和與餘項證兩個級數收斂。","en":"Prove convergence of two series with exact partial sums and remainders."},"mko_ids":["mko-infinite-series"]},{"id":"derivation-mko-parametric-statistical-model-01","title":{"zh-Hant":"兩個參數值、四個樣本點的完整模型","en":"A complete model with two parameter values and four sample points"},"summary":{"zh-Hant":"逐參數驗law並區分likelihood與posterior。","en":"Validate every parameter law and distinguish likelihood from posterior."},"mko_ids":["mko-parametric-statistical-model"]},{"id":"derivation-mko-connected-topological-space-01","title":{"zh-Hant":"排除全部非平凡開閉集","en":"Excluding every nontrivial clopen set"},"summary":{"zh-Hant":"判斷三點鏈狀拓撲是否連通。","en":"Decide connectedness of a three-point chain topology."},"mko_ids":["mko-connected-topological-space"]},{"id":"derivation-mko-maximum-matching-01","title":{"zh-Hant":"三邊圖的全部八個matching候選","en":"All eight edge-subset candidates on a three-edge graph"},"summary":{"zh-Hant":"完整2³枚舉求最大matching與全部witness。","en":"Enumerate all2³ subsets for the maximum matching and all witnesses."},"mko_ids":["mko-maximum-matching"]},{"id":"derivation-mko-modular-inverse-01","title":{"zh-Hant":"17模43的正規化逆元","en":"The normalized inverse of 17 modulo 43"},"summary":{"zh-Hant":"17模43的唯一逆類為38；17·38=15·43+1。","en":"The unique inverse residue class of 17 modulo 43 is represented by 38; 17*38=15*43+1."},"mko_ids":["mko-modular-inverse"]},{"id":"derivation-mko-rank-nullity-theorem-01","title":{"zh-Hant":"三個來源座標的秩零度","en":"Rank and nullity for three source coordinates"},"summary":{"zh-Hant":"T:R³→R²，A=[[1,2,3],[2,4,6]]有rank1、nullity2，3=1+2。","en":"For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2."},"mko_ids":["mko-rank-nullity-theorem"]},{"id":"derivation-mko-uniform-convergence-of-functions-01","title":{"zh-Hant":"三點域的精確上確界誤差","en":"Exact supremum error on a three-point domain"},"summary":{"zh-Hant":"證 g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) 一致趨零。","en":"Prove g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) converges uniformly to zero."},"mko_ids":["mko-uniform-convergence-of-functions"]}],"title":"概念推導導讀"}