{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","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":"10"},"kind":"guides","total":299,"pages":13,"page":10,"entries":[{"id":"derivation-mko-euler-theorem-number-theory-01","title":{"zh-Hant":"模10的完整unit排列","en":"The full unit permutation modulo 10"},"summary":{"zh-Hant":"對a=−3,n=10，φ(10)=4且a⁴≡1 mod10。","en":"For a=-3,n=10, phi(10)=4 and a^4=1 modulo 10."},"mko_ids":["mko-euler-theorem-number-theory"]},{"id":"derivation-mko-subsequence-01","title":{"zh-Hant":"仿射索引的全稱子數列證書","en":"An all-index subsequence certificate for affine indices"},"summary":{"zh-Hant":"由 a_n=3n-2、k(j)=1+2j 構造並再取子數列。","en":"Construct a subsequence of a_n=3n-2 using k(j)=1+2j, then take a further subsequence."},"mko_ids":["mko-subsequence"]},{"id":"derivation-mko-finite-group-orbit-01","title":{"zh-Hant":"完整orbits與partition","en":"Complete orbits and their partition"},"summary":{"zh-Hant":"X的全部互異orbits為{0,1,2}與{3}，同orbit關係是等價關係。","en":"The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation."},"mko_ids":["mko-finite-group-orbit"]},{"id":"derivation-mko-monotone-sequence-convergence-theorem-01","title":{"zh-Hant":"有界單調尾與上確界論證","en":"A bounded monotone tail and the supremum argument"},"summary":{"zh-Hant":"對 a_n=3-2/(n+1) 證單調、有界、sup=3 及極限。","en":"For a_n=3-2/(n+1), prove monotonicity, boundedness, supremum 3 and convergence."},"mko_ids":["mko-monotone-sequence-convergence-theorem"]},{"id":"derivation-mko-subspace-topology-01","title":{"zh-Hant":"相對開集不等於母空間開集","en":"Relative openness differs from ambient openness"},"summary":{"zh-Hant":"計算 Y={b,c} 的子空間拓撲與全部相對閉集。","en":"Compute the subspace topology and all relative closed sets of Y={b,c}."},"mko_ids":["mko-subspace-topology"]},{"id":"derivation-mko-chebyshev-inequality-01","title":{"zh-Hant":"平方偏差的Chebyshev推導","en":"Chebyshev from squared deviations"},"summary":{"zh-Hant":"由Markov證Chebyshev並算exact例。","en":"Derive Chebyshev from Markov and compute an exact instance."},"mko_ids":["mko-chebyshev-inequality"]},{"id":"derivation-mko-inclusion-exclusion-principle-01","title":{"zh-Hant":"三集合全部七個交集項","en":"All seven intersection terms for three sets"},"summary":{"zh-Hant":"完整核交替和與每元素的重數。","en":"Audit the full alternating sum and each element's multiplicity."},"mko_ids":["mko-inclusion-exclusion-principle"]},{"id":"derivation-mko-unique-prime-factorization-01","title":{"zh-Hant":"存在、Euclid引理與唯一質因數","en":"Existence, Euclid's lemma and unique factors"},"summary":{"zh-Hant":"正整數n>1存在唯一質因數多重集合；360=2³·3²·5，負號與units另列。","en":"Every positive integer n>1 has a unique prime multiset; 360=2^3*3^2*5, with signs and units separate."},"mko_ids":["mko-unique-prime-factorization"]},{"id":"derivation-mko-finite-directed-graph-01","title":{"zh-Hant":"方向、全部degree與walk核對","en":"Orientation, all degrees and a walk audit"},"summary":{"zh-Hant":"按loop-free simple convention核完整digraph。","en":"Audit a fully specified digraph under the declared loop-free simple convention."},"mko_ids":["mko-finite-directed-graph"]},{"id":"derivation-mko-finite-group-stabilizer-01","title":{"zh-Hant":"point0的完整固定群元素","en":"All group elements fixing point zero"},"summary":{"zh-Hant":"Stab_G(0)={e,s}為子群；Stab_G(3)=G，與orbit及action kernel不同。","en":"Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions."},"mko_ids":["mko-finite-group-stabilizer"]},{"id":"derivation-mko-bolzano-weierstrass-theorem-real-01","title":{"zh-Hant":"振盪尾的兩個收斂子數列","en":"Two convergent subsequences of an oscillatory tail"},"summary":{"zh-Hant":"對 a_n=(-1)^n+1/(n+1) 給有界性與兩個完整收斂子數列。","en":"For a_n=(-1)^n+1/(n+1), provide boundedness and two complete convergent subsequences."},"mko_ids":["mko-bolzano-weierstrass-theorem-real"]},{"id":"derivation-mko-compact-subset-01","title":{"zh-Hant":"有限子空間的緊緻性","en":"Compactness of a finite subspace"},"summary":{"zh-Hant":"證 K={b,c} 在指定母空間中緊緻，並核相對開覆蓋。","en":"Prove K={b,c} compact in its ambient space and audit a relative-open cover."},"mko_ids":["mko-compact-subset"]},{"id":"derivation-mko-law-of-total-expectation-01","title":{"zh-Hant":"含非空零質量塊的全期望","en":"Total expectation with a nonempty null block"},"summary":{"zh-Hant":"完整核partition並以兩條路計算E[X]。","en":"Audit the partition and compute E[X] by two routes."},"mko_ids":["mko-law-of-total-expectation"]},{"id":"derivation-mko-binomial-theorem-01","title":{"zh-Hant":"選取y因子的完整二項展開","en":"A full binomial expansion by choosing y factors"},"summary":{"zh-Hant":"證一般係數責任並完整展開n=4。","en":"Establish coefficient counting and expand n=4 completely."},"mko_ids":["mko-binomial-theorem"]},{"id":"derivation-mko-mobius-function-01","title":{"zh-Hant":"完整因數分解與μ(30)","en":"Complete factorization and mu(30)"},"summary":{"zh-Hant":"μ(30)=−1，μ(12)=0，μ(1)=1；30的全部divisor值加總為0。","en":"mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero."},"mko_ids":["mko-mobius-function"]},{"id":"derivation-mko-context-free-grammar-01","title":{"zh-Hant":"CFG的完整production與推導witness","en":"A CFG production list and derivation witness"},"summary":{"zh-Hant":"grammar S→aSb或ε具有aaabbb與ε的有限推導witness。","en":"The grammar S to aSb or epsilon has finite derivation witnesses for aaabbb and epsilon."},"mko_ids":["mko-context-free-grammar"]},{"id":"derivation-mko-orbit-stabilizer-theorem-01","title":{"zh-Hant":"fibers確為stabilizer左陪集","en":"Fibers as left cosets of the stabilizer"},"summary":{"zh-Hant":"point0的orbit大小3、stabilizer大小2；三個fiber均為左陪集，所以6=3·2。","en":"At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2."},"mko_ids":["mko-orbit-stabilizer-theorem"]},{"id":"derivation-mko-mean-value-theorem-01","title":{"zh-Hant":"二次多項式的均值點","en":"The mean-value point of a quadratic"},"summary":{"zh-Hant":"對 f(t)=A t^2+B t+C 在 a<b 上直接構造均值點。","en":"Directly construct a mean-value point for f(t)=A*t^2+B*t+C on a<b."},"mko_ids":["mko-mean-value-theorem"]},{"id":"derivation-mko-continuous-image-of-compact-01","title":{"zh-Hant":"拉回覆蓋再推回索引","en":"Pull back covers and return the same indices"},"summary":{"zh-Hant":"以明示有限映射呈現緊緻像的覆蓋論證。","en":"Demonstrate the compact-image cover proof with an explicit finite map."},"mko_ids":["mko-continuous-image-of-compact"]},{"id":"derivation-mko-law-of-total-variance-01","title":{"zh-Hant":"within與between的精確分解","en":"Exact within-between variance decomposition"},"summary":{"zh-Hant":"用含null塊的完整law分解variance。","en":"Decompose variance in a complete law with a null block."},"mko_ids":["mko-law-of-total-variance"]},{"id":"derivation-mko-recurrence-relation-01","title":{"zh-Hant":"oldest-to-newest二階遞迴trace","en":"An oldest-to-newest order-two recurrence trace"},"summary":{"zh-Hant":"對齊state、係數與全部初值，生成a5並證唯一性責任。","en":"Align states, coefficients and all initial values, generate a5 and explain uniqueness."},"mko_ids":["mko-recurrence-relation"]},{"id":"derivation-mko-mobius-inversion-finite-divisors-01","title":{"zh-Hant":"仿射函數在divisor poset的反演","en":"Divisor-poset inversion of an affine function"},"summary":{"zh-Hant":"f(k)=2k+1、n=6時，F(6)=28且反演得到f(6)=13；n=1亦roundtrip。","en":"For f(k)=2k+1 at n=6, F(6)=28 and inversion returns f(6)=13; n=1 roundtrips too."},"mko_ids":["mko-mobius-inversion-finite-divisors"]},{"id":"derivation-mko-pushdown-automaton-01","title":{"zh-Hant":"bottom-first PDA的aabb接受run","en":"A bottom-first PDA run accepting aabb"},"summary":{"zh-Hant":"指定PDA有讀完aabb到final f的有限run，末stack仍為[Z]。","en":"The specified PDA has a finite run consuming aabb and ending at final f with stack [Z]."},"mko_ids":["mko-pushdown-automaton"]},{"id":"derivation-mko-lagrange-theorem-finite-groups-01","title":{"zh-Hant":"全部左陪集的Lagrange計數","en":"Lagrange counting from all left cosets"},"summary":{"zh-Hant":"H={e,s}有三個互異左陪集，每個大小2，所以6=[G:H]|H|=3·2。","en":"H={e,s} has three distinct left cosets of size 2, giving 6=[G:H]|H|=3*2."},"mko_ids":["mko-lagrange-theorem-finite-groups"]}],"title":"概念推導導讀"}