{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"paths","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":{},"path":{"id":"path-pulse-foundations-20261114","title":{"zh-Hant":"Wave6第八層：kernel、商群、共軛類、標準常態、MSE、LP標準形與頂點覆蓋","en":"Wave6 Eighth Layer: Kernels, Quotient Groups, Conjugacy, Standard Normal, MSE, LP Standard Form and Vertex Covers"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-normal-subgroup","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quotient-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-action","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-orbit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-continuous-real-distribution-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-improper-integral","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-exponential-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-estimator-bias","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-estimator-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sampling-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-programming","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-homomorphism-kernel","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。","en":"For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets."}},{"id":"mko-finite-quotient-group","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。","en":"Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives."}},{"id":"mko-finite-group-conjugacy-class","position":4,"requirement":"optional","rationale":{"zh-Hant":"有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。","en":"A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group."}},{"id":"mko-standard-normal-distribution","position":5,"requirement":"optional","rationale":{"zh-Hant":"標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。","en":"The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1."}},{"id":"mko-estimator-mean-squared-error","position":6,"requirement":"optional","rationale":{"zh-Hant":"scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。","en":"For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately."}},{"id":"mko-linear-program-standard-form","position":7,"requirement":"optional","rationale":{"zh-Hant":"本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。","en":"This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0."}},{"id":"mko-vertex-cover","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。","en":"A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C."}}]},"title":"Wave6第八層：kernel、商群、共軛類、標準常態、MSE、LP標準形與頂點覆蓋"}