{"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":"11"},"kind":"knowledge","total":302,"pages":13,"page":11,"entries":[{"id":"mko-mobius-inversion-finite-divisors","title":{"zh-Hant":"有限因數格上的Möbius反演","en":"Mobius inversion on finite divisor sets"},"summary":{"zh-Hant":"若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。","en":"If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n)."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["有限因數格上的Möbius反演","AI候選","待形式化"]},{"id":"mko-pushdown-automaton","title":{"zh-Hant":"下推自動機","en":"Pushdown automaton"},"summary":{"zh-Hant":"nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。","en":"A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["下推自動機","AI候選","待形式化"]},{"id":"mko-lagrange-theorem-finite-groups","title":{"zh-Hant":"有限群的Lagrange定理","en":"Lagrange theorem for finite groups"},"summary":{"zh-Hant":"有限群G的子群H以等大左陪集分割G，因此|G|=[G:H]|H|且|H|整除|G|。","en":"Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["有限群的Lagrange定理","AI候選","待形式化"]},{"id":"mko-higher-order-derivative","title":{"zh-Hant":"高階導數","en":"Higher-order derivative"},"summary":{"zh-Hant":"以f^(0)=f及f^(n+1)=(f^(n))'遞迴定義每個存在的有限階導數。","en":"Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["高階導數","AI候選","待形式化"]},{"id":"mko-hausdorff-space","title":{"zh-Hant":"Hausdorff空間","en":"Hausdorff space"},"summary":{"zh-Hant":"拓撲空間X若每對不同點都有不交open neighborhoods即為Hausdorff。","en":"A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["Hausdorff空間","AI候選","待形式化"]},{"id":"mko-continuous-real-distribution-function","title":{"zh-Hant":"連續實值分布函數","en":"Continuous real distribution function"},"summary":{"zh-Hant":"本節點以CDF本身連續定義continuous distribution；CDF仍須nondecreasing、right-continuous且兩端極限0與1。","en":"Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["連續實值分布函數","AI候選","待形式化"]},{"id":"mko-catalan-numbers","title":{"zh-Hant":"Catalan數","en":"Catalan numbers"},"summary":{"zh-Hant":"Catalan數採C_0=1，並以convolution recurrence或兩個exact closed forms定義同一zero-based sequence。","en":"With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["Catalan數","AI候選","待形式化"]},{"id":"mko-quadratic-residue","title":{"zh-Hant":"二次剩餘","en":"Quadratic residue"},"summary":{"zh-Hant":"本節點採odd prime p上的nonzero convention：a mod p非零且等於某個nonzero square時為quadratic residue。","en":"Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["二次剩餘","AI候選","待形式化"]},{"id":"mko-nondeterministic-finite-automaton","title":{"zh-Hant":"非確定有限自動機","en":"Nondeterministic finite automaton"},"summary":{"zh-Hant":"epsilon-NFA以狀態、input symbol或epsilon對應可能後繼集合，存在一條full-input final path即接受。","en":"An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["非確定有限自動機","AI候選","待形式化"]},{"id":"mko-cauchy-theorem-finite-groups","title":{"zh-Hant":"有限群的Cauchy定理","en":"Cauchy theorem for finite groups"},"summary":{"zh-Hant":"若質數p整除有限群G的階，則G含有一個exact order為p的元素。","en":"If a prime p divides the order of a finite group G, G contains an element of exact order p."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["有限群的Cauchy定理","AI候選","待形式化"]},{"id":"mko-taylor-polynomial","title":{"zh-Hant":"Taylor多項式","en":"Taylor polynomial"},"summary":{"zh-Hant":"在中心a由f的零階到n階導數值依factorial縮放組成n階Taylor多項式。","en":"At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["Taylor多項式","AI候選","待形式化"]},{"id":"mko-compact-subset-closed-in-hausdorff","title":{"zh-Hant":"Hausdorff空間中的緊緻集為閉集","en":"A compact subset of a Hausdorff space is closed"},"summary":{"zh-Hant":"若K是Hausdorff空間X的緊緻子集，則K在X中為閉集。","en":"If K is compact in the Hausdorff space X, then K is closed in X."},"type":"theorem","domain":"topology","formalization":"not_formalized","tags":["Hausdorff空間中的緊緻集為閉集","AI候選","待形式化"]},{"id":"mko-improper-integral","title":{"zh-Hant":"瑕積分","en":"Improper integral"},"summary":{"zh-Hant":"無界區間或含奇點的積分以適當有限區間積分的one-sided limits定義，且每個必要分段須獨立收斂。","en":"Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["瑕積分","AI候選","待形式化"]},{"id":"mko-power-series-radius-of-convergence","title":{"zh-Hant":"冪級數收斂半徑","en":"Radius of convergence of a power series"},"summary":{"zh-Hant":"冪級數在中心c周圍具有內部絕對收斂、外部發散的半徑R，邊界點須逐案判定。","en":"Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["冪級數收斂半徑","AI候選","待形式化"]},{"id":"mko-legendre-symbol","title":{"zh-Hant":"Legendre符號","en":"Legendre symbol"},"summary":{"zh-Hant":"對奇質數p，Legendre符號以0、1、-1分別編碼zero class、nonzero quadratic residue與nonresidue。","en":"For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["Legendre符號","AI候選","待形式化"]},{"id":"mko-subset-construction-nfa-to-dfa","title":{"zh-Hant":"NFA到DFA的子集合構造","en":"Subset construction from an NFA to a DFA"},"summary":{"zh-Hant":"以epsilon-NFA狀態子集合為DFA狀態，start、transition與acceptance按epsilon closure定義即可保持語言。","en":"Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language."},"type":"theorem","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["NFA到DFA的子集合構造","AI候選","待形式化"]},{"id":"mko-finite-normal-subgroup","title":{"zh-Hant":"有限正規子群","en":"Finite normal subgroup"},"summary":{"zh-Hant":"有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。","en":"A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限正規子群","AI候選","待形式化"]},{"id":"mko-real-exponential-function","title":{"zh-Hant":"實指數函數","en":"Real exponential function"},"summary":{"zh-Hant":"實指數函數以對每個實數絕對收斂的冪級數E(x)=sum_{n≥0}x^n/n!定義，並滿足E(x+y)=E(x)E(y)與E(x)>0。","en":"Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實指數函數","AI候選","待形式化"]},{"id":"mko-analytic-ordinary-generating-function","title":{"zh-Hant":"解析普通生成函數","en":"Analytic ordinary generating function"},"summary":{"zh-Hant":"數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。","en":"An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["解析普通生成函數","AI候選","待形式化"]},{"id":"mko-euler-criterion-quadratic-residue","title":{"zh-Hant":"Euler二次剩餘判準","en":"Euler criterion for quadratic residues"},"summary":{"zh-Hant":"若p為奇質數且p不整除a，則a^((p-1)/2)模p等於Legendre符號(a/p)的模p代表。","en":"For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p)."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Euler二次剩餘判準","AI候選","待形式化"]},{"id":"mko-turing-machine-decision-model","title":{"zh-Hant":"Turing機判定模型","en":"Turing-machine decision model"},"summary":{"zh-Hant":"固定deterministic single-tape Turing機語義；decider必須對每個合法輸入停機並給正確accept/reject答案。","en":"Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["Turing機判定模型","AI候選","待形式化"]},{"id":"mko-gradient-vector","title":{"zh-Hant":"梯度向量","en":"Gradient vector"},"summary":{"zh-Hant":"在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。","en":"In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["梯度向量","AI候選","待形式化"]},{"id":"mko-finite-flow-network","title":{"zh-Hant":"有限流網路","en":"Finite flow network"},"summary":{"zh-Hant":"有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。","en":"A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["有限流網路","AI候選","待形式化"]},{"id":"mko-finite-group-homomorphism-kernel","title":{"zh-Hant":"有限群同態的核","en":"Kernel of a finite group homomorphism"},"summary":{"zh-Hant":"有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。","en":"For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群同態的核","AI候選","待形式化"]}],"title":"知識物件"}