{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"knowledge","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":"knowledge","total":302,"pages":13,"page":12,"entries":[{"id":"mko-finite-quotient-group","title":{"zh-Hant":"有限商群","en":"Finite quotient group"},"summary":{"zh-Hant":"有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。","en":"Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限商群","AI候選","待形式化"]},{"id":"mko-finite-group-conjugacy-class","title":{"zh-Hant":"有限群共軛類","en":"Conjugacy class in a finite group"},"summary":{"zh-Hant":"有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。","en":"A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群共軛類","AI候選","待形式化"]},{"id":"mko-standard-normal-distribution","title":{"zh-Hant":"標準常態分布","en":"Standard normal distribution"},"summary":{"zh-Hant":"標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。","en":"The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["標準常態分布","AI候選","待形式化"]},{"id":"mko-estimator-mean-squared-error","title":{"zh-Hant":"估計量均方誤差","en":"Mean squared error of an estimator"},"summary":{"zh-Hant":"scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。","en":"For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["估計量均方誤差","AI候選","待形式化"]},{"id":"mko-linear-program-standard-form","title":{"zh-Hant":"線性規劃標準形","en":"Standard form of a linear program"},"summary":{"zh-Hant":"本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。","en":"This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["線性規劃標準形","AI候選","待形式化"]},{"id":"mko-vertex-cover","title":{"zh-Hant":"頂點覆蓋","en":"Vertex cover"},"summary":{"zh-Hant":"有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。","en":"A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["頂點覆蓋","AI候選","待形式化"]},{"id":"mko-first-isomorphism-theorem-finite-groups","title":{"zh-Hant":"有限群第一同構定理","en":"First isomorphism theorem for finite groups"},"summary":{"zh-Hant":"有限群同態f:G→Q由kernel cosets誘導出G/ker f到im f的群同構，而非自動同構到整個codomain。","en":"A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["有限群第一同構定理","AI候選","待形式化"]},{"id":"mko-finite-group-center","title":{"zh-Hant":"有限群中心","en":"Center of a finite group"},"summary":{"zh-Hant":"有限群G的中心Z(G)由與每個群元素交換的全部元素構成，且是G的正規子群。","en":"The center consists of all elements commuting with every group element and is a normal subgroup."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群中心","AI候選","待形式化"]},{"id":"mko-bias-variance-decomposition","title":{"zh-Hant":"偏誤－變異數分解","en":"Bias-variance decomposition"},"summary":{"zh-Hant":"逐參數theta，scalar estimator的MSE等於其variance加上bias平方。","en":"For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["偏誤－變異數分解","AI候選","待形式化"]},{"id":"mko-linear-program-duality","title":{"zh-Hant":"線性規劃對偶","en":"Linear-programming duality"},"summary":{"zh-Hant":"對固定primal max c^Tx、Ax≤b、x≥0，其dual為min b^Ty、A^Ty≥c、y≥0；feasible pair滿足弱對偶。","en":"For primal maximize c^T*x with A*x≤b,x≥0, the dual minimizes b^T*y with A^T*y≥c,y≥0; feasible pairs satisfy weak duality."},"type":"theorem","domain":"optimization_operations_research","formalization":"not_formalized","tags":["線性規劃對偶","AI候選","待形式化"]},{"id":"mko-max-flow-min-cut-theorem","title":{"zh-Hant":"最大流最小割定理","en":"Max-flow min-cut theorem"},"summary":{"zh-Hant":"有限容量網路的最大feasible flow value等於全部s–t cuts中的最小capacity。","en":"In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts."},"type":"theorem","domain":"optimization_operations_research","formalization":"not_formalized","tags":["最大流最小割定理","AI候選","待形式化"]},{"id":"mko-minimum-vertex-cover","title":{"zh-Hant":"最小頂點覆蓋","en":"Minimum vertex cover"},"summary":{"zh-Hant":"有限simple graph中，cardinality在全部vertex covers裡最小的候選是minimum vertex cover。","en":"A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["最小頂點覆蓋","AI候選","待形式化"]},{"id":"mko-many-one-reduction","title":{"zh-Hant":"多對一歸約","en":"Many-one reduction"},"summary":{"zh-Hant":"A≤mB表示存在total computable f，使每個x皆滿足x∈A恰當f(x)∈B。","en":"A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["多對一歸約","AI候選","待形式化"]},{"id":"mko-contraction-mapping-theorem","title":{"zh-Hant":"壓縮映射定理","en":"Contraction mapping theorem"},"summary":{"zh-Hant":"非空完備度量空間上的self-map若以固定q<1壓縮所有距離，則有唯一fixed point且Picard orbit收斂。","en":"A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["壓縮映射定理","AI候選","待形式化"]},{"id":"mko-intermediate-value-theorem","title":{"zh-Hant":"介值定理","en":"Intermediate value theorem"},"summary":{"zh-Hant":"連續實函數在閉區間端點值之間取到每個中間值。","en":"A continuous real function on a closed interval attains each value between its endpoint values."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["介值定理","AI候選","待形式化"]},{"id":"mko-taylor-remainder-theorem","title":{"zh-Hant":"Taylor餘項定理","en":"Taylor remainder theorem"},"summary":{"zh-Hant":"足夠高階可微函數的Taylor approximation error由Lagrange remainder與derivative bound控制。","en":"Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["Taylor餘項定理","AI候選","待形式化"]},{"id":"mko-finite-maximum-likelihood-estimator","title":{"zh-Hant":"有限模型最大似然估計量","en":"Maximum-likelihood estimator in a finite model"},"summary":{"zh-Hant":"有限參數模型中，MLE選取使觀察likelihood最大的參數，tie需固定且與theta無關。","en":"A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型最大似然估計量","AI候選","待形式化"]},{"id":"mko-finite-sufficient-statistic","title":{"zh-Hant":"有限模型充分統計量","en":"Sufficient statistic in a finite model"},"summary":{"zh-Hant":"finite model中，給定statistic value後sample conditional law不依theta時，T對theta充分。","en":"A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型充分統計量","AI候選","待形式化"]},{"id":"mko-topological-ordering-dag","title":{"zh-Hant":"有向無環圖的拓撲排序","en":"Topological ordering of a directed acyclic graph"},"summary":{"zh-Hant":"有限有向圖存在topological ordering恰當其無directed cycle。","en":"A finite directed graph has a topological ordering exactly when it has no directed cycle."},"type":"theorem","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["有向無環圖的拓撲排序","AI候選","待形式化"]},{"id":"mko-multiplicative-order-mod-n","title":{"zh-Hant":"模n乘法階","en":"Multiplicative order modulo n"},"summary":{"zh-Hant":"gcd(a,n)=1時，ord_n(a)是a^k≡1 mod n的最小正整數k。","en":"For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["模n乘法階","AI候選","待形式化"]},{"id":"mko-series-comparison-test","title":{"zh-Hant":"級數比較判別法","en":"Comparison tests for series"},"summary":{"zh-Hant":"非負項級數可由逐項或最終逐項上界與下界傳遞收斂或發散。","en":"Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["級數比較判別法","AI候選","待形式化"]},{"id":"mko-ratio-test-series","title":{"zh-Hant":"級數比值判別法","en":"Ratio test for series"},"summary":{"zh-Hant":"非零項絕對值相鄰比極限小於一時絕對收斂，大於一時發散，等於一時不判定。","en":"For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["級數比值判別法","AI候選","待形式化"]},{"id":"mko-uniform-limit-continuity-theorem","title":{"zh-Hant":"一致極限保持連續","en":"A uniform limit of continuous functions is continuous"},"summary":{"zh-Hant":"同一domain上的連續函數列若一致收斂至f，則極限函數f連續。","en":"Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["一致極限保持連續","AI候選","待形式化"]},{"id":"mko-finite-product-topology","title":{"zh-Hant":"有限乘積拓撲","en":"Finite product topology"},"summary":{"zh-Hant":"有限多個拓撲空間的Cartesian product由open rectangles生成乘積拓撲。","en":"Open rectangles generate the topology of a finite Cartesian product of topological spaces."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["有限乘積拓撲","AI候選","待形式化"]}],"title":"知識物件"}