{"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":"13"},"kind":"knowledge","total":302,"pages":13,"page":13,"entries":[{"id":"mko-quotient-topology","title":{"zh-Hant":"商拓撲","en":"Quotient topology"},"summary":{"zh-Hant":"滿射q:X→Y使U⊆Y恰在q逆像開時為開，從而賦予Y商拓撲。","en":"For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["商拓撲","AI候選","待形式化"]},{"id":"mko-path-connected-space","title":{"zh-Hant":"道路連通空間","en":"Path-connected space"},"summary":{"zh-Hant":"空間中任意兩點若可由單位區間的連續映射連接，則空間道路連通。","en":"A space is path connected when every pair of its points is joined by a continuous map from the unit interval."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["道路連通空間","AI候選","待形式化"]},{"id":"mko-topological-connected-component","title":{"zh-Hant":"拓撲連通分支","en":"Connected component of a topological space"},"summary":{"zh-Hant":"指定點的連通分支是包含該點的極大連通子集；所有分支分割空間且各自閉。","en":"The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲連通分支","AI候選","待形式化"]},{"id":"mko-central-limit-theorem-finite-variance-iid","title":{"zh-Hant":"有限變異數IID中央極限定理","en":"Central limit theorem for finite-variance IID sequences"},"summary":{"zh-Hant":"共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。","en":"An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["有限變異數IID中央極限定理","AI候選","待形式化"]},{"id":"mko-strong-law-large-numbers-finite-valued-iid","title":{"zh-Hant":"有限值IID強大數律","en":"Strong law of large numbers for finite-valued IID sequences"},"summary":{"zh-Hant":"有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。","en":"The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["有限值IID強大數律","AI候選","待形式化"]},{"id":"mko-finite-confidence-interval-coverage","title":{"zh-Hant":"有限模型信賴區間覆蓋率","en":"Confidence-interval coverage in a finite model"},"summary":{"zh-Hant":"有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。","en":"Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型信賴區間覆蓋率","AI候選","待形式化"]},{"id":"mko-wilson-theorem","title":{"zh-Hant":"Wilson定理","en":"Wilson theorem"},"summary":{"zh-Hant":"整數n>1為質數，恰當(n-1)!模n同餘於-1。","en":"An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Wilson定理","AI候選","待形式化"]},{"id":"mko-p-adic-valuation-integers","title":{"zh-Hant":"整數p進賦值","en":"p-adic valuation of integers"},"summary":{"zh-Hant":"對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。","en":"For prime p and nonzero integer a, v_p(a) is the largest exponent of p dividing |a|; this node fixes v_p(0)=infinity."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["整數p進賦值","AI候選","待形式化"]},{"id":"mko-regular-expression","title":{"zh-Hant":"正規表示式","en":"Regular expression"},"summary":{"zh-Hant":"正規表示式由empty language、epsilon、alphabet symbols、union、concatenation與Kleene star遞迴建構語言。","en":"Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["正規表示式","AI候選","待形式化"]},{"id":"mko-pumping-lemma-regular-languages","title":{"zh-Hant":"正規語言Pumping引理","en":"Pumping lemma for regular languages"},"summary":{"zh-Hant":"每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。","en":"Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count."},"type":"theorem","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["正規語言Pumping引理","AI候選","待形式化"]},{"id":"mko-newton-method","title":{"zh-Hant":"Newton根求解方法","en":"Newton method for root finding"},"summary":{"zh-Hant":"在導數非零的目前點，以切線零點更新x至x-f(x)/f'(x)；收斂是另需局部假設的責任。","en":"At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses."},"type":"algorithm","domain":"numerical_computational","formalization":"not_formalized","tags":["Newton根求解方法","AI候選","待形式化"]},{"id":"mko-trapezoidal-quadrature","title":{"zh-Hant":"複合梯形數值積分","en":"Composite trapezoidal quadrature"},"summary":{"zh-Hant":"在嚴格遞增partition上逐段以端點平均乘各自寬度，得到定積分的複合梯形近似。","en":"On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral."},"type":"algorithm","domain":"numerical_computational","formalization":"not_formalized","tags":["複合梯形數值積分","AI候選","待形式化"]},{"id":"mko-gradient-descent","title":{"zh-Hant":"梯度下降法","en":"Gradient descent"},"summary":{"zh-Hant":"在固定Euclidean coordinates中以正步長沿負梯度更新；下降與收斂仍需目標及步長假設。","en":"In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses."},"type":"algorithm","domain":"optimization_operations_research","formalization":"not_formalized","tags":["梯度下降法","AI候選","待形式化"]},{"id":"mko-lagrange-multiplier-condition","title":{"zh-Hant":"等式約束Lagrange乘數必要條件","en":"Necessary Lagrange multiplier condition for equality constraints"},"summary":{"zh-Hant":"正則等式約束局部極值點的目標梯度，必為constraint gradients的線性組合。","en":"At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients."},"type":"theorem","domain":"optimization_operations_research","formalization":"not_formalized","tags":["等式約束Lagrange乘數必要條件","AI候選","待形式化"]}],"title":"知識物件"}