Table 1 Operational rules for physical dimensions and units
From: Integrating physical units into high-performance AI-driven scientific computing
Type | Operation | Condition | Result | Mathematical expression |
|---|---|---|---|---|
Dimension | + or − | Identical dimensions | Same as input dimensions | Dx ± Dy = Dx |
× | None | Component-wise addition of exponents | Dx ⋅ Dy = (x1 + y1,  …,  x7 + y7) | |
/ | None | Component-wise subtraction of exponents | Dx/Dy = (x1 − y1, …, x7 − y7) | |
∧ | Valid scalar exponent | Each dimension exponent scaled by scalar | \({{{{\bf{D}}}}}_{x}^{a}=(a{x}_{1},\ldots,a{x}_{7})\) | |
Unit | + or − | Identical dimensions | Adjusted scale factor | Ux ± Uy = adjusted_scale(Ux) |
× | None | Scale factors summed; dimensions combined multiplicatively | \({{{{\bf{U}}}}}_{x}*{{{{\bf{U}}}}}_{y}=1{0}^{{s}_{x}+{s}_{y}}\cdot {{{{\bf{D}}}}}_{x}{{{{\bf{D}}}}}_{y}\) | |
/ | None | Scale factors subtracted; dimensions combined divisively | \({{{{\bf{U}}}}}_{x}/{{{{\bf{U}}}}}_{y}=1{0}^{{s}_{x}-{s}_{y}}\cdot {{{{\bf{D}}}}}_{x}/{{{{\bf{D}}}}}_{y}\) | |
∧ | Valid scalar exponent | Scale factor multiplied by exponent; dimensions exponentiated | \({{{{\bf{U}}}}}_{x}^{a}=1{0}^{a{s}_{x}}\cdot {{{{\bf{D}}}}}_{x}^{a}\) |
