{"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":"6"},"kind":"knowledge","total":302,"pages":13,"page":6,"entries":[{"id":"mko-discrete-stochastic-process","title":{"zh-Hant":"離散隨機過程","en":"Discrete-time stochastic process"},"summary":{"zh-Hant":"離散隨機過程是共同機率空間上的離散時間隨機變數族；邊際分布不決定聯合律。","en":"A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["離散隨機過程","AI候選","待形式化"]},{"id":"mko-multivariable-real-function","title":{"zh-Hant":"多變數實函數","en":"Multivariable real-valued function"},"summary":{"zh-Hant":"多變數實函數把明示實向量域中的輸入映到實數；座標角色與定義域是函數資料。","en":"A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["多變數實函數","AI候選","待形式化"]},{"id":"mko-linear-independence-basis","title":{"zh-Hant":"線性獨立與基底","en":"Linear independence and bases"},"summary":{"zh-Hant":"線性獨立排除非平凡零和關係；基底同時線性獨立並生成已宣告空間。","en":"Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space."},"type":"concept","domain":"algebra","formalization":"not_formalized","tags":["線性獨立與基底","AI候選","待形式化"]},{"id":"mko-measurable-function","title":{"zh-Hant":"可測函數","en":"Measurable function"},"summary":{"zh-Hant":"來源與目標各固定sigma代數時，函數可測表示每個目標可測集的逆像皆在來源sigma代數。","en":"With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["可測函數","AI候選","待形式化"]},{"id":"mko-finite-markov-chain","title":{"zh-Hant":"有限狀態 Markov 鏈","en":"Finite-state Markov chain"},"summary":{"zh-Hant":"有限時間齊次Markov鏈以row-stochastic轉移核與需要時的初始律指定有限路徑機率。","en":"A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限狀態 Markov 鏈","AI候選","待形式化"]},{"id":"mko-partial-derivative","title":{"zh-Hant":"偏導數","en":"Partial derivative"},"summary":{"zh-Hant":"偏導數固定其他座標，只沿一個座標切片取h趨近零的單變數導數。","en":"A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["偏導數","AI候選","待形式化"]},{"id":"mko-finite-convex-combination","title":{"zh-Hant":"有限凸組合","en":"Finite convex combination"},"summary":{"zh-Hant":"有限凸組合用非負精確權重加權有限非空向量列，且權重總和恰為一。","en":"A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["有限凸組合","AI候選","待形式化"]},{"id":"mko-partial-differential-equation","title":{"zh-Hant":"偏微分方程","en":"Partial differential equation"},"summary":{"zh-Hant":"偏微分方程關聯多變數未知函數與一個或多個偏導數；殘差與主部分類需固定算子convention。","en":"A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["偏微分方程","AI候選","待形式化"]},{"id":"mko-convex-set","title":{"zh-Hant":"凸集合","en":"Convex set"},"summary":{"zh-Hant":"集合凸表示任兩點及每個lambda∈[0,1]形成的整條線段仍在集合內。","en":"A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["凸集合","AI候選","待形式化"]},{"id":"mko-boundary-value-problem","title":{"zh-Hant":"邊界值問題","en":"Boundary-value problem"},"summary":{"zh-Hant":"邊界值問題把定義域內的微分方程責任與邊界上的值或導數資料同時指定；候選通過不等於適定性。","en":"A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["邊界值問題","AI候選","待形式化"]},{"id":"mko-convex-function","title":{"zh-Hant":"凸函數","en":"Convex function"},"summary":{"zh-Hant":"凸域上的函數為凸，若每個兩點凸組合處的函數值不超過相同權重的端點函數值。","en":"A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["凸函數","AI候選","待形式化"]},{"id":"mko-convex-optimization-problem","title":{"zh-Hant":"凸最佳化問題","en":"Convex optimization problem"},"summary":{"zh-Hant":"凸最佳化問題在凸可行域上最小化凸目標；問題資格、可行性、求值與全域最優是不同責任。","en":"Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["凸最佳化問題","AI候選","待形式化"]},{"id":"mko-finite-abelian-group","title":{"zh-Hant":"有限阿貝爾群","en":"Finite abelian group"},"summary":{"zh-Hant":"有限群若完整Cayley表對所有元素對都滿足交換律，則為有限阿貝爾群。","en":"A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限阿貝爾群","AI候選","待形式化"]},{"id":"mko-real-normed-vector-space","title":{"zh-Hant":"實賦範向量空間","en":"Real normed vector space"},"summary":{"zh-Hant":"實向量空間配備正定、絕對齊次且滿足三角不等式的實值範數。","en":"A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實賦範向量空間","AI候選","待形式化"]},{"id":"mko-cauchy-sequence","title":{"zh-Hant":"Cauchy序列","en":"Cauchy sequence"},"summary":{"zh-Hant":"度量空間序列為Cauchy，若每個正epsilon最終所有兩項彼此距離皆小於epsilon。","en":"A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["Cauchy序列","AI候選","待形式化"]},{"id":"mko-finite-joint-distribution","title":{"zh-Hant":"有限聯合分布","en":"Finite joint distribution"},"summary":{"zh-Hant":"共同有限機率空間上多個隨機變數的值組與精確質量共同指定有限聯合分布。","en":"Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限聯合分布","AI候選","待形式化"]},{"id":"mko-covariance","title":{"zh-Hant":"協方差","en":"Covariance"},"summary":{"zh-Hant":"同一有限聯合分布下協方差為E[XY]-E[X]E[Y]；零協方差通常不推出獨立。","en":"Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["協方差","AI候選","待形式化"]},{"id":"mko-graph-isomorphism","title":{"zh-Hant":"圖同構","en":"Graph isomorphism"},"summary":{"zh-Hant":"有限簡單圖同構是頂點雙射，對所有頂點對完整保持鄰接與非鄰接。","en":"An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖同構","AI候選","待形式化"]},{"id":"mko-equivalence-relation","title":{"zh-Hant":"等價關係","en":"Equivalence relation"},"summary":{"zh-Hant":"固定載體上的關係若完整滿足反身、對稱與傳遞，則為等價關係。","en":"A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["等價關係","AI候選","待形式化"]},{"id":"mko-finite-unital-ring","title":{"zh-Hant":"有限含一環","en":"Finite unital ring"},"summary":{"zh-Hant":"有限含一環在同一完整有限載體上具加法阿貝爾群、乘法結合律、左右分配律及與零不同的乘法單位元。","en":"On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限含一環","AI候選","待形式化"]},{"id":"mko-norm-induced-metric","title":{"zh-Hant":"範數誘導度量","en":"Norm-induced metric"},"summary":{"zh-Hant":"同一實賦範向量空間上的範數以向量差的範數定義度量，度量公理由範數公理全稱推出。","en":"On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["範數誘導度量","AI候選","待形式化"]},{"id":"mko-complete-metric-space","title":{"zh-Hant":"完備度量空間","en":"Complete metric space"},"summary":{"zh-Hant":"每個Cauchy序列都收斂到空間內某點的度量空間稱為完備。","en":"A metric space is complete when every Cauchy sequence converges to a point still belonging to that space."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["完備度量空間","AI候選","待形式化"]},{"id":"mko-real-bilinear-form","title":{"zh-Hant":"實雙線性形式","en":"Real bilinear form"},"summary":{"zh-Hant":"實向量空間上的雙線性形式分別對兩個輸入保持向量加法與實純量乘法。","en":"A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實雙線性形式","AI候選","待形式化"]},{"id":"mko-general-linear-map","title":{"zh-Hant":"實向量空間間線性映射","en":"Linear map between real vector spaces"},"summary":{"zh-Hant":"一般實向量空間間映射若對所有向量與實純量保持線性組合，則為線性映射。","en":"A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實向量空間間線性映射","AI候選","待形式化"]}],"title":"Knowledge objects"}