{"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":"5"},"kind":"knowledge","total":302,"pages":13,"page":5,"entries":[{"id":"mko-numerical-approximation-error","title":{"zh-Hant":"數值近似與誤差","en":"Numerical approximation and error"},"summary":{"zh-Hant":"絕對誤差是近似值與參照值之差的絕對值；相對誤差另要求參照值非零並除以其絕對值。","en":"Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value."},"type":"definition","domain":"numerical_computational","formalization":"not_formalized","tags":["數值近似與誤差","AI候選","待形式化"]},{"id":"mko-optimization-problem","title":{"zh-Hant":"最佳化問題","en":"Optimization problem"},"summary":{"zh-Hant":"最佳化問題在明示可行集合上尋找使目標函數最小或最大的點；最優值與最優解集合需分開。","en":"An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["最佳化問題","AI候選","待形式化"]},{"id":"mko-dimensional-analysis","title":{"zh-Hant":"量綱分析","en":"Dimensional analysis"},"summary":{"zh-Hant":"量綱分析用基本量綱指數向量追蹤乘、除與冪，並核對方程兩側量綱是否一致。","en":"Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides."},"type":"concept","domain":"mathematical_physics","formalization":"not_formalized","tags":["量綱分析","AI候選","待形式化"]},{"id":"mko-discrete-exponential-growth","title":{"zh-Hant":"離散指數成長模型","en":"Discrete exponential growth model"},"summary":{"zh-Hant":"固定時間步與固定非負因子 r 的離散模型滿足 x_(n+1)=r x_n，因此 x_n=x_0 r^n。","en":"With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n."},"type":"definition","domain":"mathematical_biology","formalization":"not_formalized","tags":["離散指數成長模型","AI候選","待形式化"]},{"id":"mko-compound-interest","title":{"zh-Hant":"複利模型","en":"Compound-interest model"},"summary":{"zh-Hant":"固定每期利率 i 且無額外現金流時，餘額滿足 B_(n+1)=(1+i)B_n，故 B_n=B_0(1+i)^n。","en":"With a fixed rate i per period and no additional cash flows, balances satisfy B_(n+1)=(1+i)B_n and hence B_n=B_0(1+i)^n."},"type":"definition","domain":"economics_finance","formalization":"not_formalized","tags":["複利模型","AI候選","待形式化"]},{"id":"mko-mathematical-representation","title":{"zh-Hant":"數學表徵","en":"Mathematical representation"},"summary":{"zh-Hant":"數學表徵把同一對象的文字規則、符號式、有限表格、序對與座標資料分開記錄，並以明示語義橋核對一致性。","en":"Mathematical representation separately records verbal rules, symbolic expressions, finite tables, ordered pairs and coordinate data for the same object, using explicit semantic bridges to check agreement."},"type":"concept","domain":"mathematics_education","formalization":"not_formalized","tags":["數學表徵","AI候選","待形式化"]},{"id":"mko-diagnostic-mathematical-assessment","title":{"zh-Hant":"診斷性數學評量記錄","en":"Diagnostic mathematical assessment records"},"summary":{"zh-Hant":"診斷性數學評量記錄把題目、期望、作答與明示判定規則綁在一起，只產生可追溯的規則符合狀態。","en":"A diagnostic mathematical assessment record binds an item, its expected result, a response and an explicit judgment rule, producing only a traceable rule-agreement status."},"type":"concept","domain":"mathematics_education","formalization":"not_formalized","tags":["診斷性數學評量記錄","AI候選","待形式化"]},{"id":"mko-orbit-fixed-periodic","title":{"zh-Hant":"軌道、固定點與週期點","en":"Orbits, fixed points and periodic points"},"summary":{"zh-Hant":"反覆迭代自映射形成軌道；固定點週期為1，週期點的最小正週期須和任意返回時間分開。","en":"Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time."},"type":"concept","domain":"dynamical_systems","formalization":"not_formalized","tags":["軌道、固定點與週期點","AI候選","待形式化"]},{"id":"mko-initial-value-problem","title":{"zh-Hant":"初值問題","en":"Initial-value problem"},"summary":{"zh-Hant":"初值問題在微分方程外指定某點的函數值或導數值；候選核對與存在唯一性是不同責任。","en":"An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["初值問題","AI候選","待形式化"]},{"id":"mko-bisection-method","title":{"zh-Hant":"二分法的有限 bracket 更新","en":"Finite sign-bracket updates in the bisection method"},"summary":{"zh-Hant":"二分法在已提供的端點與中點符號資料上取中點並保留符號相反或零值的半區，有限步寬度減半。","en":"Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width."},"type":"algorithm","domain":"numerical_computational","formalization":"not_formalized","tags":["二分法的有限 bracket 更新","AI候選","待形式化"]},{"id":"mko-linear-programming","title":{"zh-Hant":"線性規劃","en":"Linear programming"},"summary":{"zh-Hant":"線性規劃在線性等式或不等式限制的可行域上最佳化線性目標函數；本輪題庫只驗完整有限候選網格。","en":"Linear programming optimizes a linear objective over a feasible domain defined by linear equalities or inequalities; this exercise slice checks only a complete explicit finite candidate grid."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["線性規劃","AI候選","待形式化"]},{"id":"mko-discrete-logistic-growth","title":{"zh-Hant":"離散 logistic 增量模型","en":"Discrete logistic increment model"},"summary":{"zh-Hant":"固定容量K與0≤r≤1時，本條目以x_(n+1)=x_n+r x_n(1-x_n/K)描述每步受限增量。","en":"For fixed capacity K and 0≤r≤1, this entry uses x_(n+1)=x_n+r*x_n*(1−x_n/K) to describe each capacity-limited discrete increment."},"type":"definition","domain":"mathematical_biology","formalization":"not_formalized","tags":["離散 logistic 增量模型","AI候選","待形式化"]},{"id":"mko-harmonic-oscillator-model","title":{"zh-Hant":"線性簡諧振子模型","en":"Linear harmonic-oscillator model"},"summary":{"zh-Hant":"無阻尼線性簡諧振子以m x''+k x=0及m,k>0定義；本輪不宣稱三角函數閉式解。","en":"The undamped linear harmonic oscillator is defined by m*x″+k*x=0 with m,k>0; this slice asserts no trigonometric closed-form solution."},"type":"definition","domain":"mathematical_physics","formalization":"not_formalized","tags":["線性簡諧振子模型","AI候選","待形式化"]},{"id":"mko-present-value-discounting","title":{"zh-Hant":"現值與折現","en":"Present value and discounting"},"summary":{"zh-Hant":"固定正累積因子與期數時，未來金額除以累積因子冪得到現值。","en":"For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value."},"type":"definition","domain":"economics_finance","formalization":"not_formalized","tags":["現值與折現","AI候選","待形式化"]},{"id":"mko-complex-number","title":{"zh-Hant":"複數","en":"Complex number"},"summary":{"zh-Hant":"複數以實數有序對(a,b)表示a+bi，並以i²=-1定義加減乘法。","en":"A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1."},"type":"definition","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["複數","AI候選","待形式化"]},{"id":"mko-real-vector-space","title":{"zh-Hant":"實向量空間","en":"Real vector space"},"summary":{"zh-Hant":"實向量空間配備向量加法與實數純量乘法，滿足加法交換群及相容的純量作用公理。","en":"A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實向量空間","AI候選","待形式化"]},{"id":"mko-finite-group","title":{"zh-Hant":"有限群","en":"Finite group"},"summary":{"zh-Hant":"有限群由有限集合與完整封閉Cayley表構成，並滿足結合律、唯一單位元及每元素逆元。","en":"A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群","AI候選","待形式化"]},{"id":"mko-metric-space","title":{"zh-Hant":"度量空間","en":"Metric space"},"summary":{"zh-Hant":"度量空間由集合與距離函數組成，距離非負、只在相同點為零、對稱且滿足三角不等式。","en":"A metric space has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["度量空間","AI候選","待形式化"]},{"id":"mko-finite-discrete-probability-distribution","title":{"zh-Hant":"有限離散機率分布","en":"Finite discrete probability distribution"},"summary":{"zh-Hant":"有限離散分布是在有限結果值上指定非負機率質量，全部質量精確和為一。","en":"A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限離散機率分布","AI候選","待形式化"]},{"id":"mko-complex-modulus-conjugate","title":{"zh-Hant":"複數模與共軛","en":"Complex modulus and conjugation"},"summary":{"zh-Hant":"複數共軛保留實部並反轉虛部；模平方為實部與虛部平方和。","en":"Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares."},"type":"concept","domain":"analysis","formalization":"not_formalized","tags":["複數模與共軛","AI候選","待形式化"]},{"id":"mko-linear-combination-span","title":{"zh-Hant":"線性組合與生成","en":"Linear combinations and span"},"summary":{"zh-Hant":"線性組合是有限個向量的純量加權和；生成空間收集全部有限線性組合。","en":"A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["線性組合與生成","AI候選","待形式化"]},{"id":"mko-subgroup-homomorphism","title":{"zh-Hant":"子群與群同態","en":"Subgroups and group homomorphisms"},"summary":{"zh-Hant":"子群在繼承運算下自身成群；群同態保持乘法，kernel與image由完整映射表導出。","en":"A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table."},"type":"concept","domain":"algebra","formalization":"not_formalized","tags":["子群與群同態","AI候選","待形式化"]},{"id":"mko-metric-convergence","title":{"zh-Hant":"度量空間中的收斂","en":"Convergence in a metric space"},"summary":{"zh-Hant":"度量空間序列收斂到x，表示每個正epsilon終有所有n≥N落入x的epsilon球。","en":"A sequence in a metric space converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["度量空間中的收斂","AI候選","待形式化"]},{"id":"mko-sigma-algebra-measure-space","title":{"zh-Hant":"σ-代數與測度空間","en":"Sigma-algebras and measure spaces"},"summary":{"zh-Hant":"σ-代數含空集並對補集與可數聯集封閉；測度非負、空集為零且可數可加。","en":"A sigma-algebra contains the empty set and is closed under complements and countable unions; a measure is nonnegative, zero on the empty set and countably additive."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["σ-代數與測度空間","AI候選","待形式化"]}],"title":"知識物件"}