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Step 1: Identifying key architectural elements for the gateway.
- Incorporate design elements such as pillars, arches, decorative motifs, and institutional symbols.
- Maintain symmetry and well-balanced proportions for a grand entrance.
Step 2: Sketching the gateway.
- Use proper scaling to ensure visual harmony.
- Integrate details like columns, engravings, or university emblems.
- Enhance aesthetics with decorative lamps, intricate patterns, or contemporary signage.
Step 3: Enhancing realism in the sketch.
- Apply shading techniques to create depth and dimension.
- Use texture rendering to represent materials like stone, wood, or metal effectively.
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is: