{"schema_version":"ocme-site-results-v1","locale":"en","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":"9"},"kind":"knowledge","total":302,"pages":13,"page":9,"entries":[{"id":"mko-modular-inverse","title":{"zh-Hant":"模反元素","en":"Modular inverse"},"summary":{"zh-Hant":"n≥2時，a模n有乘法反元素恰當gcd(a,n)=1；反元素在模n意義唯一。","en":"For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["模反元素","AI候選","待形式化"]},{"id":"mko-rank-nullity-theorem","title":{"zh-Hant":"秩－零度定理","en":"Rank-nullity theorem"},"summary":{"zh-Hant":"有限維線性映射T:V→W滿足dim V=dim ker T+rank T。","en":"A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T)."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["秩－零度定理","AI候選","待形式化"]},{"id":"mko-uniform-convergence-of-functions","title":{"zh-Hant":"函數列一致收斂","en":"Uniform convergence of functions"},"summary":{"zh-Hant":"一致收斂要求同一N(epsilon)同時控制domain全部x。","en":"Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["函數列一致收斂","AI候選","待形式化"]},{"id":"mko-absolute-convergence-of-series","title":{"zh-Hant":"級數絕對收斂","en":"Absolute convergence of series"},"summary":{"zh-Hant":"sum |a_n|收斂時sum a_n絕對收斂；絕對收斂推出收斂，反向不一般成立。","en":"A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["級數絕對收斂","AI候選","待形式化"]},{"id":"mko-statistical-estimator","title":{"zh-Hant":"統計估計量","en":"Statistical estimator"},"summary":{"zh-Hant":"估計量是固定且不依未知theta的可測total statistic；觀察後數值是estimate。","en":"An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["統計估計量","AI候選","待形式化"]},{"id":"mko-chinese-remainder-theorem","title":{"zh-Hant":"中國剩餘定理","en":"Chinese remainder theorem"},"summary":{"zh-Hant":"正兩兩互質模數的同餘系統在模乘積意義下存在唯一解。","en":"For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["中國剩餘定理","AI候選","待形式化"]},{"id":"mko-spanning-tree","title":{"zh-Hant":"生成樹","en":"Spanning tree"},"summary":{"zh-Hant":"生成樹是包含父圖全部頂點的連通無圈子圖。","en":"A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["生成樹","AI候選","待形式化"]},{"id":"mko-lp-convergence","title":{"zh-Hant":"Lp收斂","en":"Lp convergence"},"summary":{"zh-Hant":"共同機率空間上固定整數p≥1時，E|X_n−X|^p→0稱Lp收斂。","en":"On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["Lp收斂","AI候選","待形式化"]},{"id":"mko-direct-sum","title":{"zh-Hant":"子空間直和","en":"Direct sum of subspaces"},"summary":{"zh-Hant":"二個子空間之和若每個向量有唯一分解便是內直和；要宣稱覆蓋母空間還須另證生成。","en":"The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["子空間直和","AI候選","待形式化"]},{"id":"mko-dual-vector-space","title":{"zh-Hant":"對偶向量空間","en":"Dual vector space"},"summary":{"zh-Hant":"實向量空間的對偶空間由所有實值線性泛函組成；座標covector表示依賴基底。","en":"The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["對偶向量空間","AI候選","待形式化"]},{"id":"mko-euclidean-algorithm","title":{"zh-Hant":"Euclidean演算法","en":"Euclidean algorithm"},"summary":{"zh-Hant":"Euclidean演算法以精確帶餘除法、gcd invariant與嚴格餘數下降求最大公因數。","en":"The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders."},"type":"algorithm","domain":"number_theory","formalization":"not_formalized","tags":["Euclidean演算法","AI候選","待形式化"]},{"id":"mko-eulerian-trail","title":{"zh-Hant":"Euler trail與circuit","en":"Euler trails and circuits"},"summary":{"zh-Hant":"Euler trail恰用每條邊一次；非零邊支撐連通且奇度數為0或2時分別有circuit或open trail。","en":"An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["Euler trail與circuit","AI候選","待形式化"]},{"id":"mko-homeomorphism","title":{"zh-Hant":"同胚","en":"Homeomorphism"},"summary":{"zh-Hant":"同胚是雙射、連續且逆映射亦連續的映射；一般continuous bijection仍可能不是同胚。","en":"A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["同胚","AI候選","待形式化"]},{"id":"mko-sampling-distribution","title":{"zh-Hant":"抽樣分布","en":"Sampling distribution"},"summary":{"zh-Hant":"固定theta時，估計量把樣本分布pushforward成統計量的抽樣分布；全部theta形成一族laws。","en":"At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["抽樣分布","AI候選","待形式化"]},{"id":"mko-weak-law-large-numbers","title":{"zh-Hant":"弱大數律","en":"Weak law of large numbers"},"summary":{"zh-Hant":"IID且具有有限variance時，Chebyshev界給樣本平均依機率收斂到共同mean。","en":"For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["弱大數律","AI候選","待形式化"]},{"id":"mko-chromatic-number","title":{"zh-Hant":"色數","en":"Chromatic number"},"summary":{"zh-Hant":"有限簡單圖的色數是proper頂點著色所需顏色數的最小值，需同時具備上界與下界證書。","en":"The chromatic number of a finite simple graph is the minimum number of colors in a proper vertex coloring; equality needs both upper and lower certificates."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["色數","AI候選","待形式化"]},{"id":"mko-clique-and-independent-set","title":{"zh-Hant":"Clique與獨立集","en":"Cliques and independent sets"},"summary":{"zh-Hant":"Clique內每對不同頂點相鄰，獨立集內每對不同頂點不相鄰；最大與極大不可混同。","en":"Every pair of distinct vertices of a clique is adjacent; every pair of an independent set is nonadjacent. Inclusion-maximal and maximum-cardinality are different."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["Clique與獨立集","AI候選","待形式化"]},{"id":"mko-estimator-bias","title":{"zh-Hant":"估計量偏誤","en":"Estimator bias"},"summary":{"zh-Hant":"估計量在theta的偏誤是其theta抽樣分布期望減去目標參數；無偏要求對所有theta成立。","en":"Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["估計量偏誤","AI候選","待形式化"]},{"id":"mko-estimator-variance","title":{"zh-Hant":"估計量變異數","en":"Estimator variance"},"summary":{"zh-Hant":"估計量變異數逐theta衡量sampling law相對自身期望的二次離差，不等同偏誤或MSE。","en":"At each theta, estimator variance is the second central moment of its sampling law around its own mean; it is not bias or MSE by itself."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["估計量變異數","AI候選","待形式化"]},{"id":"mko-fermat-little-theorem","title":{"zh-Hant":"Fermat小定理","en":"Fermat's little theorem"},"summary":{"zh-Hant":"p為質數時a^p同餘a；若p不整除a則可化為a^(p-1)同餘1，逆命題不成立。","en":"For a prime p, a^p is congruent to a modulo p; if p does not divide a, a^(p−1) is congruent to1 modulo p. The converse fails."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Fermat小定理","AI候選","待形式化"]},{"id":"mko-linear-diophantine-equation","title":{"zh-Hant":"線性Diophantine方程","en":"Linear Diophantine equation"},"summary":{"zh-Hant":"非退化整數方程ax+by=c有解恰當gcd(a,b)整除c，全部解由一個整數參數給出。","en":"A nondegenerate integer equation a*x+b*y=c has a solution exactly when gcd(a,b) divides c; all solutions have one integer parameter."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["線性Diophantine方程","AI候選","待形式化"]},{"id":"mko-quotient-vector-space","title":{"zh-Hant":"商向量空間","en":"Quotient vector space"},"summary":{"zh-Hant":"V依子空間W的cosets形成商向量空間，運算須與代表元選擇無關。","en":"Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["商向量空間","AI候選","待形式化"]},{"id":"mko-finite-group-action","title":{"zh-Hant":"有限群作用","en":"Finite group action"},"summary":{"zh-Hant":"有限群在有限集合上的作用是滿足單位與乘法相容律的total action table。","en":"An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群作用","AI候選","待形式化"]},{"id":"mko-supremum-and-infimum","title":{"zh-Hant":"上確界與下確界","en":"Supremum and infimum"},"summary":{"zh-Hant":"非空且有界實數集合的supremum是最小上界，infimum是最大下界，兩者未必屬於集合。","en":"For a nonempty appropriately bounded real set, the supremum is its least upper bound and the infimum its greatest lower bound; either need not belong to the set."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["上確界與下確界","AI候選","待形式化"]}],"title":"Knowledge objects"}