{"schema_version":"ocme-site-results-v1","locale":"en","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-20261113","title":{"zh-Hant":"Wave6第七層：正規子群、實指數、解析OGF、Euler判準、Turing判定、梯度與流網路","en":"Wave6 Seventh Layer: Normal Subgroups, Real Exponential, Analytic OGFs, Euler Criterion, Turing Decisions, Gradients and Flow Networks"},"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-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-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-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-infinite-series","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-number","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-sequence-limit","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-series-radius-of-convergence","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-factorial","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-absolute-convergence-of-series","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-binomial-theorem","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-recurrence-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-sequence","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-euler-theorem-number-theory","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-quadratic-residue","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-legendre-symbol","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-computational-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-decision-problem","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-automaton","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-algorithm-correctness","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-partial-derivative","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-vector","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-multivariable-real-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-finite-directed-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-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-finite-normal-subgroup","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。","en":"A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset."}},{"id":"mko-real-exponential-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"實指數函數以對每個實數絕對收斂的冪級數E(x)=sum_{n≥0}x^n/n!定義，並滿足E(x+y)=E(x)E(y)與E(x)>0。","en":"Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0."}},{"id":"mko-analytic-ordinary-generating-function","position":4,"requirement":"optional","rationale":{"zh-Hant":"數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。","en":"An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain."}},{"id":"mko-euler-criterion-quadratic-residue","position":5,"requirement":"optional","rationale":{"zh-Hant":"若p為奇質數且p不整除a，則a^((p-1)/2)模p等於Legendre符號(a/p)的模p代表。","en":"For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p)."}},{"id":"mko-turing-machine-decision-model","position":6,"requirement":"optional","rationale":{"zh-Hant":"固定deterministic single-tape Turing機語義；decider必須對每個合法輸入停機並給正確accept/reject答案。","en":"Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer."}},{"id":"mko-gradient-vector","position":7,"requirement":"optional","rationale":{"zh-Hant":"在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。","en":"In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector."}},{"id":"mko-finite-flow-network","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。","en":"A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink."}}]},"title":"Wave6 Seventh Layer: Normal Subgroups, Real Exponential, Analytic OGFs, Euler Criterion, Turing Decisions, Gradients and Flow Networks"}