
Based on biomolecules theory and the structure of the named compounds:
\(\alpha\)-Glucose and \(\alpha\)-Galactose are epimers, differing in configuration around one specific carbon atom.
\(\alpha\)-Glucose and \(\beta\)-Glucose are anomers, differing at the anomeric carbon.
\(\alpha\)-Glucose and \(\alpha\)-Fructose are functional isomers, as they have different functional groups.
\(\alpha\)-Glucose and \(\alpha\)-Ribose are from different homologous series.

For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \[ \lim_{x \to 0} \frac{x^2 \sin(\alpha x) + (\gamma - 1)e^{x^2}}{\sin(2x - \beta x)} = 3, \] then \( \beta + \gamma - \alpha \) is equal to:

In the first configuration (1) as shown in the figure, four identical charges \( q_0 \) are kept at the corners A, B, C and D of square of side length \( a \). In the second configuration (2), the same charges are shifted to mid points C, E, H, and F of the square. If \( K = \frac{1}{4\pi \epsilon_0} \), the difference between the potential energies of configuration (2) and (1) is given by:
If \( S \) and \( S' \) are the foci of the ellipse \[ \frac{x^2}{18} + \frac{y^2}{9} = 1 \] and \( P \) is a point on the ellipse, then \[ \min (SP \cdot S'P) + \max (SP \cdot S'P) \] is equal to: