{"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":"7"},"kind":"knowledge","total":302,"pages":13,"page":7,"entries":[{"id":"mko-event-independence","title":{"zh-Hant":"事件獨立","en":"Event independence"},"summary":{"zh-Hant":"兩事件獨立以交集機率等於邊際機率乘積定義；此定義不依賴零機率條件式。","en":"Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["事件獨立","AI候選","待形式化"]},{"id":"mko-graph-walk-simple-path","title":{"zh-Hant":"圖walk與simple path","en":"Graph walks and simple paths"},"summary":{"zh-Hant":"有限無向簡單圖的walk要求每對連續頂點相鄰；simple path另要求所有頂點互異。","en":"In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖walk與simple path","AI候選","待形式化"]},{"id":"mko-finite-commutative-ring","title":{"zh-Hant":"有限交換環","en":"Finite commutative ring"},"summary":{"zh-Hant":"有限含一環若乘法對全部元素對交換，則為有限交換環；零因子仍可能存在。","en":"A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限交換環","AI候選","待形式化"]},{"id":"mko-finite-two-sided-ideal","title":{"zh-Hant":"有限雙邊理想","en":"Finite two-sided ideal"},"summary":{"zh-Hant":"有限環子集若為加法子群且吸收父環的左右乘法，則為雙邊理想。","en":"A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限雙邊理想","AI候選","待形式化"]},{"id":"mko-real-banach-space","title":{"zh-Hant":"實Banach空間","en":"Real Banach space"},"summary":{"zh-Hant":"在自身範數誘導度量下完備的實賦範向量空間稱為實Banach空間。","en":"A real normed vector space is a Banach space when it is complete in the metric induced by its own norm."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實Banach空間","AI候選","待形式化"]},{"id":"mko-real-inner-product-space","title":{"zh-Hant":"實內積空間","en":"Real inner-product space"},"summary":{"zh-Hant":"對稱且正定的實雙線性形式構成實內積；半正定或抽樣為正不足。","en":"A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實內積空間","AI候選","待形式化"]},{"id":"mko-bounded-linear-operator","title":{"zh-Hant":"有界線性算子","en":"Bounded linear operator"},"summary":{"zh-Hant":"賦範空間間線性映射若有單一非負常數統一控制所有輸出範數，則為有界線性算子。","en":"A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["有界線性算子","AI候選","待形式化"]},{"id":"mko-random-variable-independence","title":{"zh-Hant":"隨機變數獨立","en":"Independence of random variables"},"summary":{"zh-Hant":"同一有限joint law下，隨機變數獨立要求全部值對的joint mass分解為邊際質量乘積。","en":"Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["隨機變數獨立","AI候選","待形式化"]},{"id":"mko-connected-component","title":{"zh-Hant":"連通分量","en":"Connected component"},"summary":{"zh-Hant":"有限無向簡單圖中，由存在路徑所定義的頂點等價類恰為連通分量。","en":"In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["連通分量","AI候選","待形式化"]},{"id":"mko-finite-integral-domain","title":{"zh-Hant":"有限整環","en":"Finite integral domain"},"summary":{"zh-Hant":"有限交換含一環若1不等於0且沒有非零零因子，則為有限整環。","en":"A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限整環","AI候選","待形式化"]},{"id":"mko-polynomial-ring-over-finite-commutative-ring","title":{"zh-Hant":"有限交換環上的多項式環","en":"Polynomial ring over a finite commutative ring"},"summary":{"zh-Hant":"係數取自有限交換環的普通多項式以有限支撐係數序列表示，但所有任意有限次多項式合成的載體仍是無限。","en":"Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限交換環上的多項式環","AI候選","待形式化"]},{"id":"mko-inner-product-induced-norm","title":{"zh-Hant":"內積誘導範數","en":"Inner-product-induced norm"},"summary":{"zh-Hant":"同一實內積以向量自內積的非負平方根定義範數；三角不等式依Cauchy–Schwarz全稱證書。","en":"On the same real inner-product space, the norm is the nonnegative square root of the self-inner-product; the triangle inequality uses the universal Cauchy–Schwarz certificate."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["內積誘導範數","AI候選","待形式化"]},{"id":"mko-conditional-expectation","title":{"zh-Hant":"有限條件期望","en":"Finite conditional expectation"},"summary":{"zh-Hant":"有限機率空間中，相對分割生成sigma代數的條件期望在每個正質量區塊取加權平均並保持區塊積分。","en":"Relative to a partition-generated sigma-algebra on a finite probability space, conditional expectation takes the weighted mean on each positive-mass block and preserves block integrals."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限條件期望","AI候選","待形式化"]},{"id":"mko-finite-filtration","title":{"zh-Hant":"有限filtration","en":"Finite filtration"},"summary":{"zh-Hant":"有限時間filtration是在同一機率空間上隨時間資訊遞增的sigma代數序列。","en":"A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限filtration","AI候選","待形式化"]},{"id":"mko-iid-sequence","title":{"zh-Hant":"獨立同分布序列","en":"Independent identically distributed sequence"},"summary":{"zh-Hant":"IID序列同時要求每個變數有相同分布，且每個有限子族滿足mutual independence。","en":"An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["獨立同分布序列","AI候選","待形式化"]},{"id":"mko-equivalence-class","title":{"zh-Hant":"等價類","en":"Equivalence class"},"summary":{"zh-Hant":"固定完整等價關係下，元素a的等價類收集全部與a相關的載體元素。","en":"For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["等價類","AI候選","待形式化"]},{"id":"mko-finite-field","title":{"zh-Hant":"有限域","en":"Finite field"},"summary":{"zh-Hant":"有限交換含一環中每個非零元素皆有乘法逆元時構成有限域。","en":"A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限域","AI候選","待形式化"]},{"id":"mko-real-hilbert-space","title":{"zh-Hant":"實Hilbert空間","en":"Real Hilbert space"},"summary":{"zh-Hant":"在同一內積誘導範數與度量下完備的實內積空間為實Hilbert空間。","en":"A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實Hilbert空間","AI候選","待形式化"]},{"id":"mko-martingale","title":{"zh-Hant":"有限Martingale","en":"Finite martingale"},"summary":{"zh-Hant":"相對同一filtration適應且可積的有限過程，若每一步下一值條件期望等於現在值a.s.，則為Martingale。","en":"An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限Martingale","AI候選","待形式化"]},{"id":"mko-quotient-set","title":{"zh-Hant":"商集","en":"Quotient set"},"summary":{"zh-Hant":"完整等價關係的全部互異等價類組成商集，並分割原載體。","en":"The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["商集","AI候選","待形式化"]},{"id":"mko-cardinality-comparison","title":{"zh-Hant":"集合基數比較","en":"Cardinality comparison of sets"},"summary":{"zh-Hant":"明示source與target間的單射、滿射、雙射分別提供有限集合基數比較證書。","en":"An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate."},"type":"concept","domain":"foundations_logic","formalization":"not_formalized","tags":["集合基數比較","AI候選","待形式化"]},{"id":"mko-cumulative-distribution-function","title":{"zh-Hant":"累積分布函數","en":"Cumulative distribution function"},"summary":{"zh-Hant":"實值隨機變數的CDF以F(t)=P(X小於等於t)定義，為單調右連續且有限離散跳躍等於點質量。","en":"For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["累積分布函數","AI候選","待形式化"]},{"id":"mko-convergence-in-probability","title":{"zh-Hant":"依機率收斂","en":"Convergence in probability"},"summary":{"zh-Hant":"X_n依機率收斂到X，若對每個epsilon>0，偏差超過epsilon的機率趨近0。","en":"X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["依機率收斂","AI候選","待形式化"]},{"id":"mko-at-most-countable-set","title":{"zh-Hant":"至多可數集","en":"At-most-countable set"},"summary":{"zh-Hant":"集合若有限或存在到自然數的明示單射，則為至多可數；不等同可數無限。","en":"A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["至多可數集","AI候選","待形式化"]}],"title":"知識物件"}