{"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":"10"},"kind":"knowledge","total":302,"pages":13,"page":10,"entries":[{"id":"mko-topological-basis","title":{"zh-Hant":"拓撲基底","en":"Topological basis"},"summary":{"zh-Hant":"拓撲基底以覆蓋carrier與逐點局部交細化兩條公理生成拓撲。","en":"A topological basis generates a topology through carrier coverage and pointwise local intersection refinement."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲基底","AI候選","待形式化"]},{"id":"mko-markov-inequality","title":{"zh-Hant":"Markov不等式","en":"Markov inequality"},"summary":{"zh-Hant":"非負且有限期望的隨機變數在正閾值上的尾機率不超過E[X]/a。","en":"For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["Markov不等式","AI候選","待形式化"]},{"id":"mko-pigeonhole-principle","title":{"zh-Hant":"鴿籠原理","en":"Pigeonhole principle"},"summary":{"zh-Hant":"有限total map的domain比codomain大時必有兩個不同元素共享像。","en":"A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value."},"type":"theorem","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["鴿籠原理","AI候選","待形式化"]},{"id":"mko-euler-theorem-number-theory","title":{"zh-Hant":"Euler定理","en":"Euler's theorem in number theory"},"summary":{"zh-Hant":"n>=2且gcd(a,n)=1時，a的phi(n)次方模n同餘於1。","en":"For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Euler定理","AI候選","待形式化"]},{"id":"mko-subsequence","title":{"zh-Hant":"子數列","en":"Subsequence"},"summary":{"zh-Hant":"子數列由嚴格遞增的自然數索引映射選取原數列項並保留次序。","en":"A subsequence selects terms by a strictly increasing natural-index map and preserves their original order."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["子數列","AI候選","待形式化"]},{"id":"mko-finite-group-orbit","title":{"zh-Hant":"有限群作用的軌道","en":"Orbit of a finite group action"},"summary":{"zh-Hant":"有限群作用下，一點被全部群元素送到的像集合形成軌道，全部軌道分割carrier。","en":"The images of one point under all group elements form its orbit; all orbits partition the point carrier."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群作用的軌道","AI候選","待形式化"]},{"id":"mko-monotone-sequence-convergence-theorem","title":{"zh-Hant":"單調有界數列收斂定理","en":"Monotone bounded sequence convergence theorem"},"summary":{"zh-Hant":"非遞減且有上界的實數列收斂至其項集合supremum；非遞增版本對偶收斂至infimum。","en":"A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["單調有界數列收斂定理","AI候選","待形式化"]},{"id":"mko-subspace-topology","title":{"zh-Hant":"子空間拓撲","en":"Subspace topology"},"summary":{"zh-Hant":"Y⊆X的相對拓撲由全部母空間open sets與Y的交集組成。","en":"For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["子空間拓撲","AI候選","待形式化"]},{"id":"mko-chebyshev-inequality","title":{"zh-Hant":"Chebyshev不等式","en":"Chebyshev inequality"},"summary":{"zh-Hant":"有限均值與變異數的隨機變數偏離均值至少epsilon的機率不超過variance/epsilon²。","en":"For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["Chebyshev不等式","AI候選","待形式化"]},{"id":"mko-inclusion-exclusion-principle","title":{"zh-Hant":"容斥原理","en":"Inclusion-exclusion principle"},"summary":{"zh-Hant":"有限indexed set family的union cardinality是全部非空index subsets交集大小的交替和。","en":"The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset."},"type":"theorem","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["容斥原理","AI候選","待形式化"]},{"id":"mko-unique-prime-factorization","title":{"zh-Hant":"整數唯一質因數分解","en":"Unique prime factorization of integers"},"summary":{"zh-Hant":"每個n>1有prime factorization且prime multiset唯一，只允許因數次序差異。","en":"Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["整數唯一質因數分解","AI候選","待形式化"]},{"id":"mko-finite-directed-graph","title":{"zh-Hant":"有限有向圖","en":"Finite directed graph"},"summary":{"zh-Hant":"有限loop-free simple digraph由有限vertices與不含loop、無平行同向重邊的ordered-pair edge set組成。","en":"Under the loop-free simple convention, a finite digraph has a finite vertex carrier and an ordered-pair edge set without loops or duplicate same-direction edges."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["有限有向圖","AI候選","待形式化"]},{"id":"mko-finite-group-stabilizer","title":{"zh-Hant":"有限群作用的穩定子群","en":"Stabilizer of a finite group action"},"summary":{"zh-Hant":"指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。","en":"The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群作用的穩定子群","AI候選","待形式化"]},{"id":"mko-bolzano-weierstrass-theorem-real","title":{"zh-Hant":"實數列Bolzano－Weierstrass定理","en":"Bolzano-Weierstrass theorem for real sequences"},"summary":{"zh-Hant":"每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。","en":"Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["實數列Bolzano－Weierstrass定理","AI候選","待形式化"]},{"id":"mko-compact-subset","title":{"zh-Hant":"緊緻子集","en":"Compact subset"},"summary":{"zh-Hant":"K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。","en":"A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["緊緻子集","AI候選","待形式化"]},{"id":"mko-law-of-total-expectation","title":{"zh-Hant":"全期望公式","en":"Law of total expectation"},"summary":{"zh-Hant":"同一有限機率空間上，完整partition各cell的conditional mean按cell probability加權後等於E[X]。","en":"On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["全期望公式","AI候選","待形式化"]},{"id":"mko-binomial-theorem","title":{"zh-Hant":"二項式定理","en":"Binomial theorem"},"summary":{"zh-Hant":"交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。","en":"With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients."},"type":"theorem","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["二項式定理","AI候選","待形式化"]},{"id":"mko-mobius-function","title":{"zh-Hant":"Möbius函數","en":"Mobius function"},"summary":{"zh-Hant":"Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。","en":"On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["Möbius函數","AI候選","待形式化"]},{"id":"mko-context-free-grammar","title":{"zh-Hant":"上下文無關文法","en":"Context-free grammar"},"summary":{"zh-Hant":"CFG由有限且不交的nonterminals、terminals、start與單一nonterminal左側的finite productions生成語言。","en":"A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["上下文無關文法","AI候選","待形式化"]},{"id":"mko-orbit-stabilizer-theorem","title":{"zh-Hant":"軌道－穩定子定理","en":"Orbit-stabilizer theorem"},"summary":{"zh-Hant":"完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。","en":"For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["軌道－穩定子定理","AI候選","待形式化"]},{"id":"mko-mean-value-theorem","title":{"zh-Hant":"微分中值定理","en":"Mean value theorem"},"summary":{"zh-Hant":"若a<b，f在[a,b]連續且在(a,b)可微，則某個內點c的導數等於端點割線斜率。","en":"If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["微分中值定理","AI候選","待形式化"]},{"id":"mko-continuous-image-of-compact","title":{"zh-Hant":"緊緻集的連續像","en":"Continuous image of a compact set"},"summary":{"zh-Hant":"若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。","en":"If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y."},"type":"theorem","domain":"topology","formalization":"not_formalized","tags":["緊緻集的連續像","AI候選","待形式化"]},{"id":"mko-law-of-total-variance","title":{"zh-Hant":"全變異數公式","en":"Law of total variance"},"summary":{"zh-Hant":"同一有限law與完整partition生成的G下，總變異數等於cell內變異數期望加cell均值變異數。","en":"For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["全變異數公式","AI候選","待形式化"]},{"id":"mko-recurrence-relation","title":{"zh-Hant":"遞迴關係","en":"Recurrence relation"},"summary":{"zh-Hant":"order-r recurrence以明示initial values與total single-valued state update逐項決定sequence。","en":"An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["遞迴關係","AI候選","待形式化"]}],"title":"知識物件"}