{"schema_version":"ocme-site-results-v1","locale":"en","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":"3"},"kind":"knowledge","total":302,"pages":13,"page":3,"entries":[{"id":"mko-system-linear-equations","title":{"zh-Hant":"二元聯立一次方程組與退化邊界","en":"Linear systems in two variables and degenerate constraints"},"summary":{"zh-Hant":"同時滿足兩個實數線性限制才是解；非零行列式給唯一解，奇異與零列須另判一致性。","en":"A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["二元聯立一次方程組與退化邊界","AI候選","待形式化"]},{"id":"mko-conditional-probability","title":{"zh-Hant":"有限模型的條件機率","en":"Conditional probability in finite models"},"summary":{"zh-Hant":"條件事件必須有正機率；有限模型中 P(A|B)=P(A∩B)/P(B)，方向不能任意交換。","en":"The conditioning event must have positive probability. In a finite model, P(A|B)=P(A∩B)/P(B), and the direction cannot be freely exchanged."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型的條件機率","AI候選","待形式化"]},{"id":"mko-injective","title":{"zh-Hant":"單射與不同輸入的辨識","en":"Injectivity and distinct inputs"},"summary":{"zh-Hant":"單射要求不同合法輸入不能有同一輸出，不要求覆蓋整個陪域。","en":"Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["單射與不同輸入的辨識","AI候選","待形式化"]},{"id":"mko-surjective","title":{"zh-Hant":"滿射與陪域覆蓋","en":"Surjectivity and codomain coverage"},"summary":{"zh-Hant":"滿射要求陪域每個元素都有合法原像；相同輸出可以有多個原像。","en":"Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["滿射與陪域覆蓋","AI候選","待形式化"]},{"id":"mko-bijective","title":{"zh-Hant":"雙射與唯一原像","en":"Bijectivity and unique preimages"},"summary":{"zh-Hant":"雙射同時單射與滿射，等價於陪域每個元素恰有一個合法原像。","en":"A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["雙射與唯一原像","AI候選","待形式化"]},{"id":"mko-inverse-function","title":{"zh-Hant":"雙射的反函數","en":"Inverse functions of bijections"},"summary":{"zh-Hant":"雙射 f:A→B 的反函數 g:B→A 須驗證兩個方向的還原，不能把交換資料當成無條件反函數。","en":"For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["雙射的反函數","AI候選","待形式化"]},{"id":"mko-random-variable","title":{"zh-Hant":"有限樣本空間的實值隨機變數","en":"Real random variables on finite sample spaces"},"summary":{"zh-Hant":"有限模型中的隨機變數是基本結果到實數的總函數；它不是基本結果，也不是權重本身。","en":"A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限樣本空間的實值隨機變數","AI候選","待形式化"]},{"id":"mko-expectation","title":{"zh-Hant":"有限模型的期望值","en":"Expectation in finite models"},"summary":{"zh-Hant":"有限期望是合法機率權重與隨機變數值的加權總和，線性性不要求獨立。","en":"Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型的期望值","AI候選","待形式化"]},{"id":"mko-variance","title":{"zh-Hant":"有限機率模型的變異數","en":"Variance in finite probability models"},"summary":{"zh-Hant":"本輪變異數為 E[(X−E[X])²]，不是 n−1 分母的樣本變異數；零權重離群值不影響它。","en":"This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限機率模型的變異數","AI候選","待形式化"]},{"id":"mko-order-relation","title":{"zh-Hant":"實數大小與順序","en":"Usual order on the real numbers"},"summary":{"zh-Hant":"在實數論域比較大小；精確有理數交叉乘積、有限極值及含端點區間是本輪切片。","en":"The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["實數大小與順序","AI候選","待形式化"]},{"id":"mko-quadratic-function","title":{"zh-Hant":"頂點式二次函數","en":"Quadratic functions in vertex form"},"summary":{"zh-Hant":"實數全論域的 q(x)=a(x−h)²+k、a≠0；頂點、全域極值與對稱點的明示切片。","en":"A nonzero-coefficient quadratic on all of R has a vertex, an axis of symmetry, and a global minimum or maximum determined by the coefficient sign."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["頂點式二次函數","AI候選","待形式化"]},{"id":"mko-rate-of-change","title":{"zh-Hant":"兩點間的平均變化率","en":"Average rate of change between distinct inputs"},"summary":{"zh-Hant":"平均變化率是輸出差除以非零輸入差，不是瞬時導數。","en":"Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["兩點間的平均變化率","AI候選","待形式化"]},{"id":"mko-distance-formula","title":{"zh-Hant":"等單位正交座標的歐氏距離","en":"Euclidean distance in an equal-unit orthonormal frame"},"summary":{"zh-Hant":"在同一等尺度正交座標系，用非負主平方根讀取二維歐氏距離。","en":"The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["等單位正交座標的歐氏距離","AI候選","待形式化"]},{"id":"mko-circle","title":{"zh-Hant":"圓的邊界與座標判定","en":"Circle boundaries and zero-radius degeneracy"},"summary":{"zh-Hant":"圓是到固定中心距離恰等於半徑的點集，不是含內部的圓盤。","en":"A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["圓的邊界與座標判定","AI候選","待形式化"]},{"id":"mko-area","title":{"zh-Hant":"座標矩形與單位格的面積","en":"Exact area of half-open rectangles and unit-cell unions"},"summary":{"zh-Hant":"先精確處理半開矩形及有限單位格聯集，含重疊、觸邊與空矩形；不冒充一般面積理論。","en":"This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["座標矩形與單位格的面積","AI候選","待形式化"]},{"id":"mko-descriptive-statistics","title":{"zh-Hant":"有限觀測列的描述統計","en":"Descriptive summaries of nonempty observation lists"},"summary":{"zh-Hant":"保留重數的非空觀測列，精確計算均值、中位數、全距與所有最大頻率值。","en":"Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限觀測列的描述統計","AI候選","待形式化"]},{"id":"mko-matrix","title":{"zh-Hant":"有限矩陣的尺寸、索引與轉置","en":"Finite matrices: dimensions, indexing, and transpose"},"summary":{"zh-Hant":"矩陣以行、列組織元素；本輪使用非空有限實矩陣及精確整數陣列切片。","en":"Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限矩陣的尺寸、索引與轉置","AI候選","待形式化"]},{"id":"mko-linear-transformation","title":{"zh-Hant":"矩陣表示的實平面線性映射","en":"Matrix-represented linear maps on the real plane"},"summary":{"zh-Hant":"先在 R²→R² 的矩陣切片驗證加法與純量相容，不把帶非零平移的仿射式叫線性映射。","en":"Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["矩陣表示的實平面線性映射","AI候選","待形式化"]},{"id":"mko-determinant","title":{"zh-Hant":"實二階方陣的行列式","en":"Determinants of real 2×2 matrices"},"summary":{"zh-Hant":"2×2實矩陣的 ad−bc、非零行列式可逆性、乘積與純量縮放切片。","en":"This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實二階方陣的行列式","AI候選","待形式化"]},{"id":"mko-eigenvalue","title":{"zh-Hant":"實二階方陣的特徵值與非零特徵向量","en":"Real eigenvalues and nonzero eigenvectors of 2×2 matrices"},"summary":{"zh-Hant":"特徵向量須非零；候選特徵空間核含零向量，實二階矩陣可沒有實特徵值。","en":"An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實二階方陣的特徵值與非零特徵向量","AI候選","待形式化"]},{"id":"mko-congruence","title":{"zh-Hant":"明示對應的非退化三角形全等","en":"Congruence of nondegenerate triangles with explicit correspondence"},"summary":{"zh-Hant":"限定正交等單位平面中的標記非退化三角形，以三組對應邊長判全等，允許鏡射。","en":"In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["明示對應的非退化三角形全等","AI候選","待形式化"]},{"id":"mko-similarity","title":{"zh-Hant":"正倍率的標記三角形相似切片","en":"Labeled triangle similarity with a specified positive scale"},"summary":{"zh-Hant":"指定由A到B的正長度倍率 k，平方長度與指定面積以 k² 縮放，不把負座標係數當負長度倍率。","en":"Specify the positive length scale k from A to B; squared lengths and a given area scale by k². A negative coordinate coefficient is not a negative length scale."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["正倍率的標記三角形相似切片","AI候選","待形式化"]},{"id":"mko-geometric-transformations","title":{"zh-Hant":"平面座標的基本等距與均勻縮放","en":"Plane-coordinate isometries and uniform scaling"},"summary":{"zh-Hant":"平移、原點逆時針四分之一圈的整數倍旋轉、指定軸鏡射是全平面等距；均勻縮放先分清零、±1與其他係數。","en":"Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["平面座標的基本等距與均勻縮放","AI候選","待形式化"]},{"id":"mko-sequence-limit","title":{"zh-Hant":"數列極限與精確尾界","en":"Sequence limits and exact tail cutoffs"},"summary":{"zh-Hant":"從0起算的實數列以任意精度、某一自然尾界及整個尾段定義極限；本輪精確處理倒數尾項模型。","en":"A zero-based real sequence converges when every precision admits a natural cutoff controlling the entire tail; this pulse treats an exact reciprocal-tail model."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["數列極限與精確尾界","AI候選","待形式化"]}],"title":"Knowledge objects"}