{"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":"9"},"kind":"guides","total":299,"pages":13,"page":9,"entries":[{"id":"derivation-mko-absolute-convergence-of-series-01","title":{"zh-Hant":"絕對收斂與交錯調和的雙證書","en":"Absolute convergence and the two certificates for alternating harmonic"},"summary":{"zh-Hant":"區分負比幾何級數的絕對收斂與交錯調和的條件收斂。","en":"Distinguish absolute convergence of a negative-ratio geometric series from conditional convergence of alternating harmonic."},"mko_ids":["mko-absolute-convergence-of-series"]},{"id":"derivation-mko-statistical-estimator-01","title":{"zh-Hant":"同一資料函數與兩個參數law","en":"One data function under two parameter laws"},"summary":{"zh-Hant":"核total estimator並計算逐theta期望。","en":"Validate a total estimator and its parameterwise expectations."},"mko_ids":["mko-statistical-estimator"]},{"id":"derivation-mko-chinese-remainder-theorem-01","title":{"zh-Hant":"三模數CRT的構造與唯一類","en":"Constructing a three-modulus CRT class"},"summary":{"zh-Hant":"x≡2 mod3、3 mod5、2 mod7的全部整數解為23+105t。","en":"All integer solutions of x=2 mod 3, x=3 mod 5, x=2 mod 7 are 23+105t."},"mko_ids":["mko-chinese-remainder-theorem"]},{"id":"derivation-mko-spanning-tree-01","title":{"zh-Hant":"方形父圖內的spanning tree","en":"A spanning tree inside a square parent graph"},"summary":{"zh-Hant":"核全頂點、選邊存在、connected及acyclic。","en":"Audit all vertices, edge membership, connectivity and acyclicity."},"mko_ids":["mko-spanning-tree"]},{"id":"derivation-mko-lp-convergence-01","title":{"zh-Hant":"有限law的L平方精確尾端","en":"An exact L-squared tail on a finite law"},"summary":{"zh-Hant":"計算二次誤差並提供任意容忍值的尾端。","en":"Compute the second error moment and give a tail for any tolerance."},"mko_ids":["mko-lp-convergence"]},{"id":"derivation-mko-direct-sum-01","title":{"zh-Hant":"R³中唯一分解與覆蓋","en":"Unique decomposition and coverage in R3"},"summary":{"zh-Hant":"R³=span{e₁}⊕span{e₂,e₃}，唯一性與覆蓋分別成立。","en":"R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified."},"mko_ids":["mko-direct-sum"]},{"id":"derivation-mko-dual-vector-space-01","title":{"zh-Hant":"雙基底與covector換基","en":"Dual basis and covector change"},"summary":{"zh-Hant":"對b₁=(1,1)、b₂=(1,−1)，φ₁=(x+y)/2、φ₂=(x−y)/2為雙基底。","en":"For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis."},"mko_ids":["mko-dual-vector-space"]},{"id":"derivation-mko-euclidean-algorithm-01","title":{"zh-Hant":"帶符號輸入的Euclid與回代","en":"Euclid and back-substitution for signed inputs"},"summary":{"zh-Hant":"gcd(−84,30)=6，且(−84)·1+30·3=6。","en":"gcd(-84,30)=6, with (-84)*1+30*3=6."},"mko_ids":["mko-euclidean-algorithm"]},{"id":"derivation-mko-eulerian-trail-01","title":{"zh-Hant":"含孤立頂點的Euler circuit","en":"An Euler circuit with an isolated vertex"},"summary":{"zh-Hant":"完整核每條邊恰一次與非孤立支撐條件。","en":"Audit every edge exactly once and the nonisolated support condition."},"mko_ids":["mko-eulerian-trail"]},{"id":"derivation-mko-homeomorphism-01","title":{"zh-Hant":"兩個有限鏈狀空間的同胚","en":"A homeomorphism between finite chain spaces"},"summary":{"zh-Hant":"用全部逆像核對重命名映射是同胚。","en":"Audit a relabeling homeomorphism using every open preimage."},"mko_ids":["mko-homeomorphism"]},{"id":"derivation-mko-sampling-distribution-01","title":{"zh-Hant":"依fiber推送的完整抽樣分布","en":"A complete fibre-pushforward sampling distribution"},"summary":{"zh-Hant":"對兩個theta完整合併U的相同值outcomes。","en":"Aggregate equal-valued outcomes of U under both theta laws."},"mko_ids":["mko-sampling-distribution"]},{"id":"derivation-mko-weak-law-large-numbers-01","title":{"zh-Hant":"公平Bernoulli平均的Chebyshev尾端","en":"A Chebyshev tail for fair Bernoulli averages"},"summary":{"zh-Hant":"由全序列IID前提導出WLLN。","en":"Derive the WLLN from a whole-sequence IID premise."},"mko_ids":["mko-weak-law-large-numbers"]},{"id":"derivation-mko-chromatic-number-01","title":{"zh-Hant":"三角形全部二色配置的排除","en":"Excluding every two-color assignment on a triangle"},"summary":{"zh-Hant":"以上界與完整低色數搜尋證chi=3。","en":"Prove chi=3 using an upper witness and complete lower-color search."},"mko_ids":["mko-chromatic-number"]},{"id":"derivation-mko-clique-and-independent-set-01","title":{"zh-Hant":"三頂點圖的全部clique與independent sets","en":"All cliques and independent sets on three vertices"},"summary":{"zh-Hant":"完整2³子集分類求omega與alpha。","en":"Classify all2³ subsets to obtain omega and alpha."},"mko_ids":["mko-clique-and-independent-set"]},{"id":"derivation-mko-estimator-bias-01","title":{"zh-Hant":"完整參數集合上的偏誤表","en":"A bias table over the whole parameter set"},"summary":{"zh-Hant":"逐theta求EthetaU-theta，區分觀察誤差。","en":"Compute EthetaU-theta at every theta, distinct from observed error."},"mko_ids":["mko-estimator-bias"]},{"id":"derivation-mko-estimator-variance-01","title":{"zh-Hant":"同law二次矩與中心矩交叉核對","en":"Same-law second and centered moment cross-checks"},"summary":{"zh-Hant":"對全部theta算variance並核對中心定義。","en":"Compute variance at all theta values and verify centered definitions."},"mko_ids":["mko-estimator-variance"]},{"id":"derivation-mko-fermat-little-theorem-01","title":{"zh-Hant":"模7的Fermat證書與合數邊界","en":"Fermat modulo 7 and a composite boundary"},"summary":{"zh-Hant":"(−2)^6≡1 mod7；倍數7只可用a^7≡a版本；通過測試不證質數。","en":"(-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality."},"mko_ids":["mko-fermat-little-theorem"]},{"id":"derivation-mko-linear-diophantine-equation-01","title":{"zh-Hant":"負係數方程的全部整數解","en":"All integer solutions with a negative coefficient"},"summary":{"zh-Hant":"−6x+9y=15全部解為x=5+3t,y=5+2t，t∈Z。","en":"All integer solutions of -6x+9y=15 are x=5+3t,y=5+2t, t in Z."},"mko_ids":["mko-linear-diophantine-equation"]},{"id":"derivation-mko-quotient-vector-space-01","title":{"zh-Hant":"對角線的商空間","en":"Quotient by the diagonal line"},"summary":{"zh-Hant":"R²/span{(1,1)}的基底為[(0,1)]，維度1；零coset是整條對角線。","en":"R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal."},"mko_ids":["mko-quotient-vector-space"]},{"id":"derivation-mko-finite-group-action-01","title":{"zh-Hant":"六元素群對四點的完整作用","en":"A complete six-element action on four points"},"summary":{"zh-Hant":"明示24格表為有效群作用，但不是free或transitive。","en":"The explicit 24-entry table is a valid group action, but it is neither free nor transitive."},"mko_ids":["mko-finite-group-action"]},{"id":"derivation-mko-supremum-and-infimum-01","title":{"zh-Hant":"有限最大值與未取到的區間上確界","en":"Finite maxima and unattained interval suprema"},"summary":{"zh-Hant":"求 A={-2,1/3,5/2} 與 B=(0,1) 的 sup/inf，逐一核界與最小性。","en":"Find sup/inf of A={-2,1/3,5/2} and B=(0,1), checking bounds and extremality."},"mko_ids":["mko-supremum-and-infimum"]},{"id":"derivation-mko-topological-basis-01","title":{"zh-Hant":"有限鏈的基底與全部生成開集","en":"A finite-chain basis and every generated open"},"summary":{"zh-Hant":"驗 B={{a},{a,b},X} 是指定拓撲的基底。","en":"Verify B={{a},{a,b},X} is a basis for the specified topology."},"mko_ids":["mko-topological-basis"]},{"id":"derivation-mko-markov-inequality-01","title":{"zh-Hant":"由逐點不等式得到Markov界","en":"Deriving a Markov bound pointwise"},"summary":{"zh-Hant":"證有限law的界並比較exact tail。","en":"Prove the finite-law bound and compare its exact tail."},"mko_ids":["mko-markov-inequality"]},{"id":"derivation-mko-pigeonhole-principle-01","title":{"zh-Hant":"完整映射的碰撞與ceil負載","en":"Collision and ceiling load in a complete map"},"summary":{"zh-Hant":"從全部fibers證碰撞及廣義下界。","en":"Use all fibres to prove a collision and the generalized lower bound."},"mko_ids":["mko-pigeonhole-principle"]}],"title":"概念推導導讀"}