{"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":"11"},"kind":"guides","total":299,"pages":13,"page":11,"entries":[{"id":"derivation-mko-higher-order-derivative-01","title":{"zh-Hant":"四次多項式的全部導數階","en":"Every derivative order of a quartic"},"summary":{"zh-Hant":"由低到高係數 [2,-3,0,5,-2] 計算所有階導數。","en":"Compute all derivative orders for low-to-high coefficients [2,-3,0,5,-2]."},"mko_ids":["mko-higher-order-derivative"]},{"id":"derivation-mko-hausdorff-space-01","title":{"zh-Hant":"三點離散空間的全部點對分離","en":"Separating every pair in a discrete three-point space"},"summary":{"zh-Hant":"完整核三點離散空間的 Hausdorff 性，並對照一個失敗點對。","en":"Exhaustively audit Hausdorffness of a discrete three-point space and contrast a failed pair."},"mko_ids":["mko-hausdorff-space"]},{"id":"derivation-mko-continuous-real-distribution-function-01","title":{"zh-Hant":"有平坦段的連續CDF","en":"A continuous CDF with a flat interval"},"summary":{"zh-Hant":"從knots完整構造CDF並核atomless性。","en":"Construct a complete knot-based CDF and audit atomlessness."},"mko_ids":["mko-continuous-real-distribution-function"]},{"id":"derivation-mko-catalan-numbers-01","title":{"zh-Hant":"Catalan卷積與整除證書","en":"Catalan convolution and a divisibility certificate"},"summary":{"zh-Hant":"從C0完整生成C4並核兩個closed forms。","en":"Generate C4 from C0 and check both closed forms."},"mko_ids":["mko-catalan-numbers"]},{"id":"derivation-mko-quadratic-residue-01","title":{"zh-Hant":"完整模7平方表","en":"The complete nonzero square table modulo 7"},"summary":{"zh-Hant":"模7非零平方剩餘為{1,2,4}；−5≡2有根3、4，3無根。","en":"Modulo 7, the nonzero quadratic residues are {1,2,4}; the class of -5 equals the class of 2 and has root classes 3 and 4, while 3 has no square root modulo 7."},"mko_ids":["mko-quadratic-residue"]},{"id":"derivation-mko-nondeterministic-finite-automaton-01","title":{"zh-Hant":"NFA的existential有限路徑","en":"An existential finite accepting NFA path"},"summary":{"zh-Hant":"指定NFA接受aab，即使另一分支dead；epsilon loop本身不接受。","en":"The specified NFA accepts aab despite a dead alternative branch; an epsilon loop alone does not accept."},"mko_ids":["mko-nondeterministic-finite-automaton"]},{"id":"derivation-mko-cauchy-theorem-finite-groups-01","title":{"zh-Hant":"prime因子的exact-order witnesses","en":"Exact-order witnesses for the prime divisors"},"summary":{"zh-Hant":"此六元素群的prime divisors2、3分別有s與r的exact-order witnesses；無order6元素。","en":"In this six-element group, prime divisors 2,3 have exact-order witnesses s,r; there is no element of order 6."},"mko_ids":["mko-cauchy-theorem-finite-groups"]},{"id":"derivation-mko-taylor-polynomial-01","title":{"zh-Hant":"中心平移與 factorial 的 Taylor 多項式","en":"A Taylor polynomial retaining the center and factorials"},"summary":{"zh-Hant":"以中心 a=1、order 2 構造指定三次多項式的 T_2。","en":"Construct T_2 at center a=1 for the specified cubic polynomial."},"mko_ids":["mko-taylor-polynomial"]},{"id":"derivation-mko-compact-subset-closed-in-hausdorff-01","title":{"zh-Hant":"緊緻把分離鄰域縮為有限個","en":"Compactness reduces separating neighborhoods to finitely many"},"summary":{"zh-Hant":"證 Hausdorff 空間中的緊緻子集 K 閉。","en":"Prove a compact subset K of a Hausdorff space is closed."},"mko_ids":["mko-compact-subset-closed-in-hausdorff"]},{"id":"derivation-mko-improper-integral-01","title":{"zh-Hant":"每個無界端與奇點分別取極限","en":"Separate limits at every infinite endpoint and singularity"},"summary":{"zh-Hant":"計算三種明示冪函數瑕積分，並區分 principal value。","en":"Compute three explicit power-law improper integrals and distinguish principal value."},"mko_ids":["mko-improper-integral"]},{"id":"derivation-mko-power-series-radius-of-convergence-01","title":{"zh-Hant":"半徑、端點與完整係數律","en":"Radius, endpoints and complete coefficient laws"},"summary":{"zh-Hant":"對幾何係數 2^n 求 R，並展示端點及 0/infinity 邊界。","en":"Find R for geometric coefficients 2^n, and exhibit endpoint and zero/infinity boundaries."},"mko_ids":["mko-power-series-radius-of-convergence"]},{"id":"derivation-mko-legendre-symbol-01","title":{"zh-Hant":"獨立平方表定義三值Legendre符號","en":"The three-valued Legendre symbol from squares"},"summary":{"zh-Hant":"模7符號對0、1、2、3、4、5、6依序為0、1、1、−1、1、−1、−1。","en":"Modulo 7 the symbol on 0,1,2,3,4,5,6 is respectively 0,1,1,-1,1,-1,-1."},"mko_ids":["mko-legendre-symbol"]},{"id":"derivation-mko-subset-construction-nfa-to-dfa-01","title":{"zh-Hant":"epsilon-NFA的完整reachable subset DFA","en":"The complete reachable-subset DFA of an epsilon-NFA"},"summary":{"zh-Hant":"完整列轉移並以prefix induction證語言等價。","en":"List every transition and prove language equivalence by prefix induction."},"mko_ids":["mko-subset-construction-nfa-to-dfa"]},{"id":"derivation-mko-finite-normal-subgroup-01","title":{"zh-Hant":"全部共軛認證rotation子群正規","en":"Normality of rotations from every conjugate"},"summary":{"zh-Hant":"N={e,r,r²}為正規子群；H={e,s}雖為子群但不正規。","en":"N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal."},"mko_ids":["mko-finite-normal-subgroup"]},{"id":"derivation-mko-real-exponential-function-01","title":{"zh-Hant":"全級數的加法律與嚴格正性","en":"Addition and strict positivity of the full exponential series"},"summary":{"zh-Hant":"從 E(x)=sum x^n/n! 證全實數絕對收斂、加法律、逆元與正性。","en":"From E(x)=sum x^n/n!, prove absolute convergence for all real x, addition, inverses and positivity."},"mko_ids":["mko-real-exponential-function"]},{"id":"derivation-mko-analytic-ordinary-generating-function-01","title":{"zh-Hant":"含初值邊界項的Fibonacci型OGF","en":"A Fibonacci-type OGF with initial boundary terms"},"summary":{"zh-Hant":"在證明收斂的域內從遞迴推出OGF。","en":"Derive the OGF from a recurrence on a justified convergence domain."},"mko_ids":["mko-analytic-ordinary-generating-function"]},{"id":"derivation-mko-euler-criterion-quadratic-residue-01","title":{"zh-Hant":"兩條獨立路徑核Euler判準","en":"Two independent routes for Euler's criterion"},"summary":{"zh-Hant":"p=7,a=−4時Legendre值−1且a³≡6≡−1 mod7；2則給+1。","en":"For p=7,a=-4 the Legendre value is -1 and a^3=6=-1 mod 7; 2 gives +1."},"mko_ids":["mko-euler-criterion-quadratic-residue"]},{"id":"derivation-mko-turing-machine-decision-model-01","title":{"zh-Hant":"二元字串偶數1的scan decider","en":"A scan decider for an even number of ones"},"summary":{"zh-Hant":"完整machine、trace、ranking與全input正確性。","en":"Give the complete machine, trace, ranking and all-input correctness."},"mko_ids":["mko-turing-machine-decision-model"]},{"id":"derivation-mko-gradient-vector-01","title":{"zh-Hant":"座標梯度、駐點與可微的區別","en":"Coordinate gradients, stationarity and differentiability"},"summary":{"zh-Hant":"求指定二元多項式梯度並核兩個常見錯誤推論。","en":"Compute the gradient of a specified bivariate polynomial and audit two false inferences."},"mko_ids":["mko-gradient-vector"]},{"id":"derivation-mko-finite-flow-network-01","title":{"zh-Hant":"逐邊與逐內點核可行流","en":"Checking every edge and internal vertex of a feasible flow"},"summary":{"zh-Hant":"核一個含反平行、零容量邊與零流量點的完整網路流值。","en":"Audit a complete flow network with antiparallel edges, a zero-capacity edge and a zero-flow vertex."},"mko_ids":["mko-finite-flow-network"]},{"id":"derivation-mko-finite-group-homomorphism-kernel-01","title":{"zh-Hant":"C₄到C₂的kernel、fibers與單射性","en":"Kernel, fibers and injectivity from C4 to C2"},"summary":{"zh-Hant":"f:C₄→C₂，f(a)=a mod2的kernel為{0,2}且正規；fibers是其cosets，f非單射。","en":"For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective."},"mko_ids":["mko-finite-group-homomorphism-kernel"]},{"id":"derivation-mko-finite-quotient-group-01","title":{"zh-Hant":"全部代表pairs的商群表","en":"A quotient table checked on every representative pair"},"summary":{"zh-Hant":"G/N的兩cosets A=N、B={s,rs,r²s}具良定乘法AA=A,AB=BA=B,BB=A。","en":"G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A."},"mko_ids":["mko-finite-quotient-group"]},{"id":"derivation-mko-finite-group-conjugacy-class-01","title":{"zh-Hant":"完整共軛作用與三個classes","en":"Full conjugation action and three classes"},"summary":{"zh-Hant":"全部共軛類是{e}、{r,r²}、{s,rs,r²s}；r與s不共軛。","en":"The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate."},"mko_ids":["mko-finite-group-conjugacy-class"]},{"id":"derivation-mko-standard-normal-distribution-01","title":{"zh-Hant":"標準常態的對稱推導與named facts","en":"Standard-normal symmetry with explicit named facts"},"summary":{"zh-Hant":"清楚分開Gaussian前提與由它們推得的CDF對稱性。","en":"Separate Gaussian prerequisites from derived CDF symmetry."},"mko_ids":["mko-standard-normal-distribution"]}],"title":"概念推導導讀"}