OCME · Bilingual mathematics & derivations
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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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.
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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.
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Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain.
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Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages.
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A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage.
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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.
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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.
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Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence.
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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.
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The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend.
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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.
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Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative.
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The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences.
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A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton.
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This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases.
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Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities.
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Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances.
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Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear.
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This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices.
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An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues.
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In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection.
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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.
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Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients.
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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.