{"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":"8"},"kind":"knowledge","total":302,"pages":13,"page":8,"entries":[{"id":"mko-bipartite-graph","title":{"zh-Hant":"二分圖","en":"Bipartite graph"},"summary":{"zh-Hant":"有限無向簡單圖若全體頂點可分為兩個互斥部分且每條邊跨部，則為二分圖。","en":"A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["二分圖","AI候選","待形式化"]},{"id":"mko-convergence-in-distribution","title":{"zh-Hant":"依分布收斂","en":"Convergence in distribution"},"summary":{"zh-Hant":"極限CDF每個連續點上F_n逐點收斂到F時，稱依分布收斂。","en":"Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["依分布收斂","AI候選","待形式化"]},{"id":"mko-finite-quotient-ring","title":{"zh-Hant":"有限環的商環","en":"Quotient ring of a finite ring"},"summary":{"zh-Hant":"有限環依proper雙邊理想的加法cosets形成商環，運算須與代表元選擇無關。","en":"Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限環的商環","AI候選","待形式化"]},{"id":"mko-graph-coloring","title":{"zh-Hant":"圖著色","en":"Graph coloring"},"summary":{"zh-Hant":"proper k-coloring使用至多k種顏色並使每條邊兩端顏色不同。","en":"A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖著色","AI候選","待形式化"]},{"id":"mko-vector-subspace","title":{"zh-Hant":"實向量子空間","en":"Real vector subspace"},"summary":{"zh-Hant":"實向量空間的非空子集若含零向量並對向量加法與每個實純量乘法封閉，則為向量子空間。","en":"A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實向量子空間","AI候選","待形式化"]},{"id":"mko-closure-and-interior","title":{"zh-Hant":"拓撲閉包與內部","en":"Topological closure and interior"},"summary":{"zh-Hant":"集合閉包是所有閉超集的交，內部是所有開子集的聯；兩者相對於指定拓撲。","en":"The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲閉包與內部","AI候選","待形式化"]},{"id":"mko-open-cover-and-subcover","title":{"zh-Hant":"開覆蓋與子覆蓋","en":"Open cover and subcover"},"summary":{"zh-Hant":"開覆蓋由指定拓撲的開集組成且聯集包含目標集合；子覆蓋須取自原覆蓋並仍覆蓋同一目標。","en":"An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["開覆蓋與子覆蓋","AI候選","待形式化"]},{"id":"mko-almost-sure-convergence","title":{"zh-Hant":"幾乎必然收斂","en":"Almost-sure convergence"},"summary":{"zh-Hant":"同一機率空間上的隨機變數序列若在零機率例外集外逐點收斂，則幾乎必然收斂。","en":"Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["幾乎必然收斂","AI候選","待形式化"]},{"id":"mko-graph-matching","title":{"zh-Hant":"圖匹配","en":"Graph matching"},"summary":{"zh-Hant":"有限圖的matching是彼此不共享端點的邊子集；空matching合法。","en":"A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖匹配","AI候選","待形式化"]},{"id":"mko-euler-totient-function","title":{"zh-Hant":"Euler φ函數","en":"Euler totient function"},"summary":{"zh-Hant":"Euler φ(n)計數一個完整剩餘代表系中與n互質的代表；約定φ(1)=1。","en":"Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["Euler φ函數","AI候選","待形式化"]},{"id":"mko-finite-automaton","title":{"zh-Hant":"確定有限自動機","en":"Deterministic finite automaton"},"summary":{"zh-Hant":"DFA由有限狀態集、有限字母表、總轉移函數、唯一初始狀態與接受狀態子集組成。","en":"A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["確定有限自動機","AI候選","待形式化"]},{"id":"mko-vector-space-dimension","title":{"zh-Hant":"有限維向量空間的維度","en":"Dimension of a finite-dimensional vector space"},"summary":{"zh-Hant":"有限維向量空間的維度是任一有限基底的元素數；基底不同但基數不變。","en":"The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限維向量空間的維度","AI候選","待形式化"]},{"id":"mko-linear-map-kernel","title":{"zh-Hant":"線性映射的核","en":"Kernel of a linear map"},"summary":{"zh-Hant":"線性映射T的核是全部滿足T(x)=0的來源向量，並形成來源子空間。","en":"The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["線性映射的核","AI候選","待形式化"]},{"id":"mko-linear-map-image","title":{"zh-Hant":"線性映射的像","en":"Image of a linear map"},"summary":{"zh-Hant":"線性映射的像是所有輸出T(x)的集合；矩陣映射的像等於column span。","en":"The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["線性映射的像","AI候選","待形式化"]},{"id":"mko-topological-boundary","title":{"zh-Hant":"拓撲邊界","en":"Topological boundary"},"summary":{"zh-Hant":"集合A的拓撲邊界為closure(A)減去interior(A)，亦等於closure(A)交closure(complement A)。","en":"The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲邊界","AI候選","待形式化"]},{"id":"mko-compact-topological-space","title":{"zh-Hant":"緊緻拓撲空間","en":"Compact topological space"},"summary":{"zh-Hant":"拓撲空間緊緻，若每個開覆蓋都有有限子覆蓋；每個有限拓撲空間皆緊緻。","en":"A space is compact when every open cover has a finite subcover; every finite topological space is compact."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["緊緻拓撲空間","AI候選","待形式化"]},{"id":"mko-bayes-theorem","title":{"zh-Hant":"貝氏定理","en":"Bayes theorem"},"summary":{"zh-Hant":"正分母條件下，Bayes定理以共同intersection與priors/likelihoods精確反轉條件機率。","en":"With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["貝氏定理","AI候選","待形式化"]},{"id":"mko-regular-language","title":{"zh-Hant":"正規語言","en":"Regular language"},"summary":{"zh-Hant":"正規語言是存在一個完整DFA可辨識其全部字串的語言。","en":"A language is regular when a complete DFA recognizes all and exactly its words."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["正規語言","AI候選","待形式化"]},{"id":"mko-matrix-rank","title":{"zh-Hant":"矩陣秩","en":"Matrix rank"},"summary":{"zh-Hant":"矩陣秩是row/column space維度，可由精確rank-preserving消去的pivot數計算。","en":"Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["矩陣秩","AI候選","待形式化"]},{"id":"mko-pointwise-convergence-of-functions","title":{"zh-Hant":"函數列逐點收斂","en":"Pointwise convergence of functions"},"summary":{"zh-Hant":"同一domain上的函數列逐點收斂，若每個固定x的值序列收斂至f(x)，且N可依x與epsilon改變。","en":"A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["函數列逐點收斂","AI候選","待形式化"]},{"id":"mko-infinite-series","title":{"zh-Hant":"無窮級數","en":"Infinite series"},"summary":{"zh-Hant":"無窮級數以partial-sum sequence的極限定義；項趨零必要但不充分。","en":"An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["無窮級數","AI候選","待形式化"]},{"id":"mko-parametric-statistical-model","title":{"zh-Hant":"參數統計模型","en":"Parametric statistical model"},"summary":{"zh-Hant":"參數統計模型為每個theta指定同一有限可測sample space上的完整機率law P_theta。","en":"A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["參數統計模型","AI候選","待形式化"]},{"id":"mko-connected-topological-space","title":{"zh-Hant":"連通拓撲空間","en":"Connected topological space"},"summary":{"zh-Hant":"拓撲空間連通，若不存在由兩個不交非空開集組成的separation；等價於無非平凡clopen subset。","en":"A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["連通拓撲空間","AI候選","待形式化"]},{"id":"mko-maximum-matching","title":{"zh-Hant":"最大匹配","en":"Maximum matching"},"summary":{"zh-Hant":"最大匹配是在完整有限圖全部合法matching中邊數最大者；可有多個且不必perfect。","en":"A maximum matching has the greatest edge count among all valid matchings of the complete finite graph; it may be nonunique and need not be perfect."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["最大匹配","AI候選","待形式化"]}],"title":"知識物件"}