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❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ❆ ❘ ❆ ■ ◆ ● ❊ ◆ ■ ❊ ❘ ❖ ❙ ❈ ❖ ◆        

❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙      

❱ ❡ ✐ ♥ ✶ ✳ ✶  

■ ♥ ❣ ✳ ▼ ✳ ❙ ❝ ✳ ❆ ♥ ♦ ♥ ✐ ♦ ▼ ❡ ❞ ❛ ♥ ♦  

✶ ❞ ❡ ❛ ❜ ✐ ❧ ❞ ❡ ✷ ✵ ✶ ✺  

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❮ ♥ ❞ ✐ ❝ ❡ ❣ ❡ ♥ ❡ ❛ ❧  

■ ✳ ❆ ❙ ❊ ❈ ❚ ❖ ❙ ❇ ➪ ❙ ■ ❈ ❖ ❙ ✺  

✶ ✳ ■ ◆ ❚ ❘ ❖ ❉ ❯ ❈ ❈ ■ ❖ ◆ ✻  

✷ ✳ ❈ ❖ ◆ ❈ ❊ ❚ ❖ ❙ ❇ ➪ ❙ ■ ❈ ❖ ❙ ❉ ❊ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ✽  

✷ ✳ ✶ ✳ ❚ ✐ ♣ ♦ ❞ ❡ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽  

✷ ✳ ✷ ✳ ❉ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ✉ ♥ ▼ ♦ ❞ ❡ ❧ ♦ ♣ ❛ ❛ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾  

✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙ ✶ ✷  

✸ ✳ ✶ ✳ ■ ♥ ♦ ❞ ✉ ❝ ❝ ✐ ♥ ❛ ❧ ❙ ♦ ❢ ✇ ❛ ❡ ▲ ✐ ❜ ❡ ❞ ❡ ❈ ❞ ✐ ❣ ♦ ❆ ❜ ✐ ❡ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✷  

✸ ✳ ✷ ✳ ❙ ♦ ❢ ✇ ❛ ❡ ❛ ✉ ✐ ❧ ✐ ③ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸  

✸ ✳ ✷ ✳ ✶ ✳ ▼ ❛ ① ✐ ♠ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸  

✸ ✳ ✷ ✳ ✷ ✳ ❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ❈ ❛ ❧ ❝ ✴ ▲ ✐ ❜ ❡ ❖ ✣ ❝ ❡ ❈ ❛ ❧ ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✹  

✸ ✳ ✷ ✳ ✸ ✳ ❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✺  

✸ ✳ ✷ ✳ ✹ ✳ ❙ ❛ ❣ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✻  

✸ ✳ ✷ ✳ ✺ ✳ ❖ ❝ ❛ ✈ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✻  

✸ ✳ ✷ ✳ ✻ ✳ ❙ ❝ ✐ ❧ ❛ ❜ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✽  

■ ■ ✳ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ❆ ▲ ■ ❈ ❆ ❉ ❆ ✷ ✵  

✹ ✳ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ◆ ❖ ▲ ■ ◆ ❊ ❆ ▲ ◆ ❖ ❘ ❊ ❙ ❚ ❘ ■ ◆ ● ■ ❉ ❆ ❊ ◆ ❯ ◆ ❆ ❱ ❆ ❘ ■ ❆ ❇ ▲ ❊ ✷ ✶  

✹ ✳ ✶ ✳ ❉ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ✉ ♥ ♠ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶  

✹ ✳ ✷ ✳ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ◆ ♦ ▲ ✐ ♥ ❡ ❛ ❧ ❡ ♥ ▼ ❛ ① ✐ ♠ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷  

✹ ✳ ✸ ✳ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ◆ ♦ ▲ ✐ ♥ ❡ ❛ ❧ ❡ ♥ ❙ ❛ ❣ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✼  

✹ ✳ ✸ ✳ ✶ ✳ ❈ ❛ ♦ ❊ ♣ ❡ ❝ ✐ ❛ ❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶  

✹ ✳ ✹ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✹  

✺ ✳ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ▲ ■ ◆ ❊ ❆ ▲ ❘ ❊ ❙ ❚ ❘ ■ ◆ ● ■ ❉ ❆ ❊ ◆ ❱ ❆ ❘ ■ ❆ ❙ ❱ ❆ ❘ ■ ❆ ❇ ▲ ❊ ❙ ✸ ✺  

✺ ✳ ✶ ✳ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ▲ ✐ ♥ ❡ ❛ ❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✺  

✺ ✳ ✶ ✳ ✶ ✳ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ▲ ✐ ♥ ❡ ❛ ❧ ❡ ♥ ▼ ❛ ① ✐ ♠ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✽  

✺ ✳ ✶ ✳ ✷ ✳ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ▲ ✐ ♥ ❡ ❛ ❧ ❡ ♥ ❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ✴ ▲ ✐ ❜ ❡ ❖ ✣ ❝ ❡ ❈ ❛ ❧ ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶  

✺ ✳ ✶ ✳ ✸ ✳ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ▲ ✐ ♥ ❡ ❛ ❧ ❡ ♥ ❙ ❛ ❣ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✼  

✺ ✳ ✷ ✳ ▼ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❘ ❡ ❞ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✽  

✺ ✳ ✷ ✳ ✶ ✳ ▼ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❚ ❛ ♥ ♣ ♦ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✵  

✺ ✳ ✷ ✳ ✷ ✳ ▼ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❆ ✐ ❣ ♥ ❛ ❝ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✺  

✺ ✳ ✷ ✳ ✸ ✳ ▼ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❘ ✉ ❛ ▼ ❈ ♦ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✽  

✺ ✳ ✷ ✳ ✹ ✳ ▼ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❋ ❧ ✉ ❥ ♦ ▼ ① ✐ ♠ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✸  

✺ ✳ ✷ ✳ ✺ ✳ ❆ ❜ ♦ ❧ ▼ ♥ ✐ ♠ ♦ ❞ ❡ ❊ ① ♣ ❛ ♥ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✽  

✺ ✳ ✷ ✳ ✻ ✳ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ❧ ❆ ❣ ❡ ♥ ❡ ❱ ✐ ❛ ❥ ❡ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶  

✺ ✳ ✸ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸  

✺ ✳ ✸ ✳ ✶ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ▲ ✐ ♥ ❡ ❛ ❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸  

✺ ✳ ✸ ✳ ✷ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ▼ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❚ ❛ ♥ ♣ ♦ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸  

✺ ✳ ✸ ✳ ✸ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ▼ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❆ ✐ ❣ ♥ ❛ ❝ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸  

✺ ✳ ✸ ✳ ✹ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❘ ✉ ❛ ▼ ❈ ♦ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸  

✷ 

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❮ ♥ ❞ ✐ ❝ ❡ ❣ ❡ ♥ ❡ ❛ ❧  

✺ ✳ ✸ ✳ ✺ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❋ ❧ ✉ ❥ ♦ ▼ ① ✐ ♠ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸  

✺ ✳ ✸ ✳ ✻ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❆ ❜ ♦ ❧ ▼ ♥ ✐ ♠ ♦ ❞ ❡ ❊ ① ♣ ❛ ♥ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸  

✻ ✳ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ◆ ❖ ▲ ■ ◆ ❊ ❆ ▲ ◆ ❖ ❘ ❊ ❙ ❚ ❘ ■ ◆ ● ■ ❉ ❆ ❊ ◆ ❱ ❆ ❘ ■ ❆ ❙ ❱ ❆ ❘ ■ ❆ ❇ ▲ ❊ ❙ ✼ ✹  

✻ ✳ ✶ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✼  

✼ ✳ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ◆ ❖ ▲ ■ ◆ ❊ ❆ ▲ ❘ ❊ ❙ ❚ ❘ ■ ◆ ● ■ ❉ ❆ ❊ ◆ ❱ ❆ ❘ ■ ❆ ❙ ❱ ❆ ❘ ■ ❆ ❇ ▲ ❊ ❙ ✼ ✽  

✼ ✳ ✶ ✳ ❘ ❊ ❙ ❚ ❘ ■ ❈ ❈ ■ ❖ ◆ ❊ ❙ ❉ ❊ ■ ● ❯ ❆ ▲ ❉ ❆ ❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✽  

✼ ✳ ✷ ✳ ❘ ❊ ❙ ❚ ❘ ■ ❈ ❈ ■ ❖ ◆ ❊ ❙ ❉ ❊ ❉ ❊ ❙ ■ ● ❯ ❆ ▲ ❉ ❆ ❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✵  

✼ ✳ ✸ ✳ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✹  

✽ ✳ ▼ ➱ ❚ ❖ ❉ ❖ ❙ ❍ ❊ ❯ ❘ ❮ ❙ ❚ ■ ❈ ❖ ❙ ❆ ❘ ❆ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆ ✽ ✺  

■ ■ ■ ✳ ▼ ❖ ❉ ❊ ▲ ❖ ❙ ❉ ❊ ▲ ■ ◆ ❊ ❆ ❙ ❉ ❊ ❊ ❙ ❊ ❘ ❆ ✽ ✻  

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❮ ♥ ❞ ✐ ❝ ❡ ❞ ❡ ✜ ❣ ✉ ❛    

✶ ✳ ✶ ✳ ▼ ❛ ♣ ❛ ❈ ♦ ♥ ❝ ❡ ♣ ✉ ❛ ❧ ❞ ❡ ❧ ❈ ♦ ♥ ❡ ♥ ✐ ❞ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼  

✷ ✳ ✶ ✳ ♦ ❝ ❡ ♦ ❞ ❡ ❙ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾  

✷ ✳ ✷ ✳ ♦ ❝ ❡ ♦ ♣ ❛ ❛ ❡ ❧ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ✉ ♥ ♠ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶  

✸ ✳ ✶ ✳ ■ ♠ ❛ ❣ ❡ ♥ ❡ ❞ ❡ ▼ ❛ ① ✐ ♠ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✹  

✸ ✳ ✷ ✳ ❙ ♦ ❧ ✈ ❡ ❞ ❡ ❈ ❛ ❧ ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✺  

✸ ✳ ✸ ✳ ■ ♠ ❛ ❣ ❡ ♥ ❞ ❡ ❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✻  

✸ ✳ ✹ ✳ ■ ♠ ❛ ❣ ❡ ♥ ❡ ❞ ❡ ❙ ❛ ❣ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✼  

✸ ✳ ✺ ✳ ■ ♠ ❛ ❣ ❡ ♥ ❡ ❞ ❡ ❖ ❝ ❛ ✈ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✽  

✸ ✳ ✻ ✳ ■ ♠ ❛ ❣ ❡ ♥ ❡ ❞ ❡ ❙ ❝ ✐ ❧ ❛ ❜ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✾  

✺ ✳ ✶ ✳ ❊ ❥ ❡ ♠ ♣ ❧ ♦ ❞ ❡ ❘ ❡ ❞ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✾  

✺ ✳ ✷ ✳ ❚ ✐ ♣ ♦ ❞ ❡ ● ❛ ❢ ♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✵  

✺ ✳ ✸ ✳ ❊ ❥ ❡ ♠ ♣ ❧ ♦ ❞ ❡ ✉ ♥ ❛ ❡ ❞ ❝ ♦ ♥ ♣ ❡ ♦ ✭ ❞ ✐ ❛ ♥ ❝ ✐ ❛ ✮ ② ✉ ❡ ♣ ❡ ❝ ✐ ✈ ❛ ♠ ❛ ✐ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✵  

✺ ✳ ✹ ✳ ❘ ✉ ❛ ▼ ❛ ❈ ♦ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✽  

✺ ✳ ✺ ✳ ❊ ❥ ❡ ♠ ♣ ❧ ♦ ❞ ❡ ❜ ♦ ❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✾  

✼ ✳ ✶ ✳ ❊ ❥ ❡ ♠ ♣ ❧ ♦ ❞ ❡ ✉ ♥ ❛ ❢ ✉ ♥ ❝ ✐ ♥ ❝ ♦ ♥ ♠ ✉ ❝ ❤ ♦ ♣ ✐ ♠ ♦ ❧ ♦ ❝ ❛ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶  

✹ 

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5/86

❛ ❡ ■ ✳  

❆ ❙ ❊ ❈ ❚ ❖ ❙ ❇ ➪ ❙ ■ ❈ ❖ ❙        

✺ 

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6/86

✶ ✳ ■ ◆ ❚ ❘ ❖ ❉ ❯ ❈ ❈ ■ ❖ ◆  

▲ ❛ ✐ ❞ ❡ ❛ ❞ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❡ ❛ ♥ ♦ ❛ ✉ ❣ ✐ ❤ ❛ ❝ ❡ ❛ ❧ ❣ ✉ ♥ ♦ ❛ ♦ ❝ ✉ ❛ ♥ ❞ ♦ ♠ ❡ ❢ ✉ ♦ ❧ ✐ ❝ ✐ ❛ ❞ ♦ ♣ ♦  

♣ ✐ ♠ ❡ ❛ ✈ ❡ ③ ✐ ♠ ♣ ❛ ✐ ❡ ❧ ❝ ✉ ♦ ❞ ❡ ♥ ♦ ♠ ✐ ♥ ❛ ❞ ♦ ❆ ❧ ❣ ❡ ❜ ❛ ② ❚ ❡ ❝ ♥ ♦ ❧ ♦ ❣ ❛ ■ ♥ ❢ ♦ ♠ ✐ ❝ ❛ ❡ ♥ ❧ ❛ ▼ ❛ ❡ ❛ ❡ ♥  

■ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ❱ ✐ ❛ ❧ ❞ ❡ ❧ ❛ ❯ ♥ ✐ ✈ ❡ ✐ ❞ ❛ ❞ ❞ ❡ ❙ ❛ ♥ ❈ ❛ ❧ ♦ ❞ ❡ ● ✉ ❛ ❡ ♠ ❛ ❧ ❛ ✳ ❊ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ❡ ❡ ❝ ✉ ♦ ✱ ❝ ♦ ♠ ♦  

♠ ❡ ❢ ✉ ♣ ❧ ❛ ♥ ❡ ❛ ❞ ♦ ❡ ♥ ❡ ❡ ♠ ♦ ♠ ❡ ♥ ♦ ✱ ❡ ❛ ✉ ❡ ❧ ♦ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ❞ ❡ ♠ ❛ ❡ ❛ ❛ ♣ ❡ ♥ ❞ ✐ ❡ ❛ ♥ ❛ ✉ ✐ ❧ ✐ ③ ❛  

❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ✐ ♥ ❢ ♦ ♠ ✐ ❝ ❛ ♣ ❛ ❛ ❧ ❛ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ❝ ♦ ♠ ♣ ❧ ❡ ❥ ♦ ✳

■ ♥ ✐ ❝ ✐ ❛ ❧ ♠ ❡ ♥ ❡ ❡ ♠ ❡ ♦ ❧ ✐ ❝ ✐ ✉ ❡ ❡ ❧ ❝ ✉ ♦ ❡ ♦ ✐ ❡ ♥ ❛ ❛ ❛ ❧ ✉ ♦ ❞ ❡ ❧ ♣ ♦ ❣ ❛ ♠ ❛ ▼ ❛ ❧ ❛ ❜  ➤ 

✶ 

❝ ♦ ♠ ♦ ❧ ❛  

❜ ❛ ❡ ❞ ❡ ❧ ❝ ✉ ♦ ✳ ▼ ❛ ❧ ❛ ❜  ➤  ❡ ❡ ❝ ♦ ♥ ♦ ❝ ✐ ❞ ♦ ❛ ♥ ✐ ✈ ❡ ❧ ♠ ✉ ❞ ✐ ❛ ❧ ❝ ♦ ♠ ♦ ❧ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❞ ❡ ❝ ❧ ❝ ✉ ❧ ♦ ♠ ❛ ❡ ✲  

♠ ✐ ❝ ♦ ♠ ❛ ✉ ✐ ❧ ✐ ③ ❛ ❞ ❛ ❡ ♥ ❝ ✐ ❡ ♥ ❝ ✐ ❛ ❡ ✐ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ✳ ❙ ✐ ♥ ❡ ♠ ❜ ❛ ❣ ♦ ✱ ♣ ♦ ❛ ❛ ❡ ❞ ❡ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ❛ ❞ ❡ ✐ ♣ ♦  

❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ✱ ✉ ✉ ♦ ✐ ♠ ♣ ❧ ✐ ❝ ❛ ❡ ❧ ♣ ❛ ❣ ♦ ❞ ❡ ✉ ♥ ❛ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ♣ ♦ ✉ ♥ ♣ ❡ ♦ ❞ ♦ ❞ ❡ ✐ ❡ ♠ ♣ ♦ ❧ ✐ ♠ ✐ ❛ ❞ ♦ ✳ ❊ ♦ ❛ ❡  

❡ ✐ ❛ ❞ ❡ ✈ ❡ ♥ ❛ ❥ ❛ ② ❛ ✉ ❡ ♣ ❛ ❛ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❡ ❧ ❝ ✉ ♦ ❡ ❡ ✉ ✐ ❡ ❡ ✉ ♥ ❧ ❛ ❜ ♦ ❛ ♦ ✐ ♦ ❡ ✉ ✐ ♣ ❛ ❞ ♦ ❝ ♦ ♥ ❡ ❧ ♣ ♦ ✲  

❣ ❛ ♠ ❛ ② ❧ ❛ ❡ ♣ ❡ ❝ ✐ ✈ ❛ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ✱ ❧ ♦ ❝ ✉ ❛ ❧ ♣ ❡ ❡ ♥ ❛ ❜ ❛ ✉ ♥ ♦ ❜ ❝ ✉ ❧ ♦ ✱ ❛ ♥ ♦ ♣ ❛ ❛ ✐ ♠ ♣ ❛ ✐ ❧ ❛ ❝ ❧ ❛ ❡  

♠ ❛ ❣ ✐ ❛ ❧ ❡ ❝ ♦ ♠ ♦ ♣ ❛ ❛ ✉ ❡ ❧ ♦ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ❧ ❧ ❡ ✈ ❛ ❛ ♥ ❛ ❝ ❛ ❜ ♦ ❧ ❛ ❡ ♣ ❡ ❝ ✐ ✈ ❛ ♣ ❝ ✐ ❝ ❛ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥  

♣ ❛ ❛ ❢ ❛ ♠ ✐ ❧ ✐ ❛ ✐ ③ ❛ ❡ ❝ ♦ ♥ ❡ ❧ ❛ ♣ ❧ ✐ ❝ ❛ ✐ ✈ ♦ ✳

❖ ♦ ❛ ♣ ❡ ❝ ♦ ✉ ❡ ✈ ❛ ❧ ❡ ❧ ❛ ♣ ❡ ♥ ❛ ♠ ❡ ♥ ❝ ✐ ♦ ♥ ❛ ✱ ❡ ❛ ❧ ❣ ♦ ✉ ❡ ❤ ❡ ♣ ♦ ❞ ✐ ❞ ♦ ♦ ❜ ❡ ✈ ❛ ② ❝ ♦ ♠ ♣ ♦ ❜ ❛ ❛ ❛ ✈  

❞ ❡ ♠ ✐ ♣ ♦ ♣ ✐ ❛ ❡ ① ♣ ❡ ✐ ❡ ♥ ❝ ✐ ❛ ✿ ❛ ❛ ❛ ♣ ❡ ♥ ❞ ❡ ❛ ✉ ✐ ❧ ✐ ③ ❛ ✉ ♥ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ✱ ❞ ❡ ❧ ✐ ♣ ♦ ✉ ❡ ❡ ❛ ✭ ♠ ❡ ❝ ♥ ✐ ❝ ❛ ✱  

❡ ❝ ♥ ♦ ❧ ❣ ✐ ❝ ❛ ✉ ♦ ♦ ✐ ♣ ♦ ✮ ❡ ♥ ❡ ❝ ❡ ❛ ✐ ♦ ❡ ♥ ❡ ✉ ♥ ♣ ♦ ♣ ✐ ♦ ✉ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ✜ ♥ ✐ ❞ ♦ ♣ ❛ ❛ ✉ ✉ ♦ ✳ ❊ ❧ ♣ ♦ ❝ ❡ ♦  

❞ ❡ ❛ ♣ ❡ ♥ ❞ ✐ ③ ❛ ❥ ❡ ❞ ❡ ❧ ♦ ♣ ♦ ❝ ❡ ❞ ✐ ♠ ✐ ❡ ♥ ♦ ② ♠ ♦ ❞ ♦ ♣ ❛ ❛ ❡ ❧ ✉ ♦ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ✐ ♥ ❢ ♦ ♠ ✐ ❝ ❛ ✱ ❝ ✉ ❛ ♥ ❞ ♦  

♥ ♦ ❡ ❞ ❞ ❡ ♥ ♦ ❞ ❡ ✉ ♥ ❝ ♦ ♥ ❡ ① ♦ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ✱ ❡ ❢ ❣ ✐ ❧ ② ♣ ♦ ❝ ♦ ❡ ✜ ❝ ❛ ③ ✱ ② ❛ ✉ ❡ ❧ ♦ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ❛ ♣ ❡ ♥ ❞ ❡ ♥  

❡ ❧ ✏ ❝ ♠ ♦ ✑ ♣ ❡ ♦ ♥ ♦ ❡ ❧ ✏ ♣ ❛ ❛ ✉ ✑ ✱ ♣ ♦ ❧ ♦ ✉ ❡ ♠ ❛ ❡ ♠ ♣ ❛ ♥ ♦ ✉ ❡ ❛ ❞ ❡ ❧ ♦ ❝ ♦ ♥ ♦ ❝ ✐ ♠ ✐ ❡ ♥ ♦ ❝ ❛ ❡ ♥ ❡ ♥ ❡ ❧  

♦ ❧ ✈ ✐ ❞ ♦ ✳ ▼ ✉ ❝ ❤ ♦ ❞ ❡ ❧ ♦ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ✉ ❡ ❝ ✉ ❛ ♥ ❧ ❛ ♠ ❛ ❡ ❛ ♦ ♥ ♣ ♦ ❢ ❡ ✐ ♦ ♥ ❛ ❧ ❡ ✉ ❡ ❝ ✉ ❧ ♠ ✐ ♥ ❛ ♦ ♥ ✉  

❡ ✉ ❞ ✐ ♦ ❛ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ✉ ❛ ❤ ❛ ❝ ❡ ② ❛ ✈ ❛ ✐ ♦ ❛ ♦ ② ♠ ✉ ❝ ❤ ♦ ❞ ❡ ✉ ❝ ♦ ♥ ❝ ❡ ♣ ♦ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ❡  

❡ ♥ ❝ ✉ ❡ ♥ ❛ ♥ ❡ ♥ ❞ ❡ ✉ ♦ ❧ ♦ ✉ ❡ ❞ ✐ ✜ ❝ ✉ ❧ ❛ ❛ ♥ ♠ ❛ ❡ ❧ ♣ ♦ ❝ ❡ ♦ ✱ ② ❛ ✉ ❡ ❡ ♥ ♠ ✉ ❝ ❤ ♦ ❝ ❛ ♦ ❤ ❛ ② ✉ ❡ ✈ ♦ ❧ ✈ ❡  

❛ ✈ ✐ ✐ ❛ ❡ ♠ ❛ ❜ ✐ ❝ ♦ ❡ ♥ ❧ ❛ ❞ ✐ ✐ ♥ ❛ ❡ ❛ ❝ ♦ ♠ ♦ ❧ ❣ ❡ ❜ ❛ ✱ ❝ ❧ ❝ ✉ ❧ ♦ ❞ ✐ ❢ ❡ ❡ ♥ ❝ ✐ ❛ ❧ ❡ ✐ ♥ ❡ ❣ ❛ ❧ ✱ ❛ ❧ ❣ ❡ ❜ ❛  

♠ ❛ ✐ ❝ ✐ ❛ ❧ ✱ ❡ ❝ ✳

❛ ❛ ✉ ❜ ❛ ♥ ❛ ❧ ❛ ♣ ✐ ♠ ❡ ❛ ❞ ✐ ✜ ❝ ✉ ❧ ❛ ❞ ✱ ❡ ❧ ❛ ❝ ✐ ♦ ♥ ❛ ❞ ❛ ❝ ♦ ♥ ❡ ❧ ✉ ♦ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ❡ ✉ ❡  

❡ ✉ ✐ ❡ ❡ ♥ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ✱ ❤ ❡ ♦ ♣ ❛ ❞ ♦ ❡ ♥ ❧ ♦ ❧ ✐ ♠ ♦ ❛ ♦ ♣ ♦ ❡ ❧ ✉ ♦ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ✉ ❡ ♣ ❡ ❡ ♥ ❡ ❝ ❡ ♥ ❛ ❧ ❛  

❝ ♦ ✐ ❡ ♥ ❡ ❞ ❡ ❧ ❙ ♦ ❢ ✇ ❛ ❡ ▲ ✐ ❜ ❡ ✭ ❋ ❖ ❙ ❙ ✱ ❋ ❡ ❡ ❛ ♥ ❞ ❖ ♣ ❡ ♥ ❙ ♦ ✉ ❝ ❡ ❙ ♦ ❢ ✇ ❛ ❡ ✱ ♣ ♦ ✉ ✐ ❣ ❧ ❛ ❡ ♥ ✐ ♥ ❣ ❧ ✮ ✳

❉ ❡ ♥ ♦ ❞ ❡ ❡ ❛ ❝ ❛ ❡ ❣ ♦ ❛ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ✉ ♥ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❞ ❡ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ✉ ❡ ♣ ♦ ❡ ❡ ♥ ♦ ❞ ❛ ♦ ❝ ❛ ✐ ♦ ❞ ❛  

❧ ❛ ❝ ❛ ❛ ❝ ❡ ✐ ❝ ❛ ❡ ✉ ❡ ✐ ❞ ❛ ♣ ❛ ❛ ❝ ✉ ♠ ♣ ❧ ✐ ❡ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ❧ ❝ ✉ ♦ ✱ ✉ ❡ ♦ ♥ ✐ ❣ ✉ ❛ ❧ ♦ ❡ ♥ ❛ ❧ ❣ ✉ ♥ ♦ ❝ ❛ ♦  

♠ ♣ ♦ ❞ ❡ ♦ ❛ ✉ ❡ ❡ ❧ ♦ ❢ ✇ ❛ ❡ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ② ✉ ❡ ✐ ❡ ♥ ❡ ♥ ❧ ❛ ❡ ♥ ♦ ♠ ❡ ✈ ❡ ♥ ❛ ❥ ❛ ❞ ❡ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ♥ ♦ ❜ ❡ ♥ ❡ ❡  

✐ ♥ ♥ ✐ ♥ ❣ ♥ ✐ ♣ ♦ ❞ ❡ ❡ ✐ ❝ ❝ ✐ ♥ ♣ ❛ ❛ ✉ ✐ ♥ ❛ ❧ ❛ ❝ ✐ ♥ ② ✉ ♦ ✱ ❡ ♥ ❢ ♦ ♠ ❛ ❣ ❛ ✉ ✐ ❛ ② ♣ ♦ ✐ ❡ ♠ ♣ ♦ ✐ ♥ ❞ ❡ ✜ ♥ ✐ ❞ ♦ ✳

❖ ♦ ❛ ♣ ❡ ❝ ♦ ✉ ❡ ❝ ♦ ♥ ✐ ❞ ❡ ♦ ✐ ♠ ♣ ♦ ❛ ♥ ❡ ❡ ❡ ❧ ♥ ♦ ❡ ✐ ♥ ❣ ✐ ❡ ❧ ❝ ✉ ♦ ❛ ✉ ♥ ❛ ♦ ❧ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ✱ ♣ ✉ ❡ ♦  

✉ ❡ ❡ ♦ ❧ ✐ ♠ ✐ ❛ ❛ ❧ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ❛ ❧ ✉ ♦ ❞ ❡ ❧ ♦ ❢ ✇ ❛ ❡ ✉ ❡ ❝ ♦ ♥ ♦ ❝ ❡ ✱ ❧ ♦ ✉ ❡ ♦ ❝ ❛ ✐ ♦ ♥ ❛ ✉ ❡ ❛ ❧ ❡ ♥ ❝ ♦ ♥ ❛ ❡ ❡ ♥  

❡ ❧ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ♣ ♦ ❢ ❡ ✐ ♦ ♥ ❛ ❧ ✱ ✐ ❞ ✐ ❝ ❤ ♦ ♣ ♦ ❣ ♠ ❛ ♥ ♦ ❡ ❞ ✐ ♣ ♦ ♥ ✐ ❜ ❧ ❡ ② ♠ ❛ ♥ ✐ ❡ ❛ ❛ ❞ ❡ ✉ ♥ ♦ ❢ ✇ ❛ ❡  

❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ❝ ✉ ② ♦ ❝ ♦ ♦ ❞ ❡ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ♠ ✐ ❡ ♥ ♦ ❡ ❡ ❧ ❡ ✈ ❛ ❞ ♦ ✱ ♥ ♦ ♣ ✉ ❡ ❞ ❛ ❡ ♥ ❡ ❧ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ♥ ❡ ❝ ❡ ❛ ✐ ❛ ♣ ❛ ❛  

❧ ❛ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ✳ ❊ ♣ ♦ ❛ ❧ ❛ ③ ♥ ✉ ❡ ❡ ♥ ❡ ❡ ❝ ✉ ♦ ❡ ♣ ❡ ❡ ♥ ❛ ♥ ✈ ❛ ✐ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛  

✉ ❡ ♣ ✉ ❡ ❞ ❡ ♥ ✉ ❛ ❡ ❡ ♥ ❞ ✐ ✐ ♥ ❛ ✐ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ✱ ❛ ❧ ❣ ✉ ♥ ❛ ❡ ♥ ♠ ❞ ❡ ✉ ♥ ❛ ✐ ✉ ❛ ❝ ✐ ♥ ✱ ❧ ♦ ✉ ❡ ❛ ❜ ❡ ❧ ❛  

♣ ♦ ✐ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ❡ ❞ ❡ ❡ ♥ ❡ ✉ ♥ ❛ ❡ ♥ ❛ ❧ ❜ ✐ ❡ ♥ ❡ ✉ ✐ ♣ ❛ ❞ ♦ ♣ ❛ ❛ ❛ ❢ ♦ ♥ ❛ ❝ ✉ ❛ ❧ ✉ ✐ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ✳

❊ ♥ ❧ ♦ ✉ ❡ ❡ ♣ ❡ ❝ ❛ ❛ ❧ ❡ ❣ ✉ ♥ ❞ ♦ ✐ ♥ ❝ ♦ ♥ ✈ ❡ ♥ ✐ ❡ ♥ ❡ ✱ ❡ ❧ ❞ ❡ ❧ ♣ ♦ ♣ ✐ ♦ ❞ ❡ ✜ ♥ ✐ ❞ ♦ ✱ ♦ ♣ ♣ ♦ ✉ ♥ ❝ ❛ ♠ ♣ ♦  

❞ ❡ ✉ ♠ ❛ ✐ ♠ ♣ ♦ ❛ ♥ ❝ ✐ ❛ ❡ ♥ ♦ ❞ ❛ ❧ ❛ ❛ ♠ ❛ ❞ ❡ ✐ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ✱ ♣ ❡ ♦ ✉ ❡ ❧ ❛ ♠ ❡ ♥ ❛ ❜ ❧ ❡ ♠ ❡ ♥ ❡ ♥ ♦ ✐ ❡ ♠ ♣ ❡  

❡ ✐ ♥ ❝ ❧ ✉ ❞ ♦ ❡ ♥ ❧ ❛ ❡ ❞ ❡ ❞ ❡ ❡ ✉ ❞ ✐ ♦ ❞ ❡ ❛ ❧ ❣ ✉ ♥ ❛ ✐ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ✿ ❧ ❛ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✳ ❊ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ♦ ❞ ♦  

✐ ♥ ❣ ❡ ♥ ✐ ❡ ♦ ✱ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ✐ ❡ ♥ ❡ ♠ ❡ ♥ ❡ ❞ ❡ ✉ ❡ ❛ ❞ ❡ ❡ ♣ ❡ ❝ ✐ ❛ ❧ ✐ ③ ❛ ❝ ✐ ♥ ✱ ❡ ❡ ♦ ❧ ✈ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❤ ❛ ❝ ✐ ❡ ♥ ❞ ♦ ✉ ♦  

♣ ✐ ♠ ♦ ❞ ❡ ❧ ♦ ❡ ❝ ✉ ♦ ❛ ✉ ❞ ✐ ♣ ♦ ✐ ❝ ✐ ♥ ✱ ♣ ♦ ❧ ♦ ✉ ❡ ❡ ❡ ❡ ♠ ❛ ❞ ❡ ❜ ❡ ❛ ❡ ❞ ❡ ✐ ♥ ❝ ❧ ✉ ✐ ♥ ♦ ❜ ❧ ✐ ❣ ❛ ❞ ❛ ❡ ♥  

✶ 

❤ ♣ ✿ ✴ ✴ ✇ ✇ ✇ ✳ ♠ ❛ ❤ ✇ ♦ ❦ ✳ ❝ ♦ ♠ ✴ ♣ ♦ ❞ ✉ ❝ ✴ ♠ ❛ ❧ ❛ ❜ ✴  

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✶ ✳ ■ ◆ ❚ ❘ ❖ ❉ ❯ ❈ ❈ ■ ❖ ◆  

❋ ✐ ❣ ✉ ❛ ✶ ✳ ✶ ✳ ✿ ▼ ❛ ♣ ❛ ❈ ♦ ♥ ❝ ❡ ♣ ✉ ❛ ❧ ❞ ❡ ❧ ❈ ♦ ♥ ❡ ♥ ✐ ❞ ♦  

♦ ❞ ♦ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❛ ❝ ❛ ❞ ♠ ✐ ❝ ♦ ❞ ❡ ✐ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ✳ ❊ ♥ ❡ ❧ ❝ ❛ ♦ ❞ ❡ ❧ ❛ ■ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ❈ ✐ ✈ ✐ ❧ ✱ ✉ ❡ ❡ ❞ ❡ ❞ ♦ ♥ ❞ ❡  

♣ ♦ ✈ ✐ ❡ ♥ ❡ ♥ ❧ ❛ ♠ ❛ ② ♦ ❛ ❞ ❡ ♣ ♦ ❢ ❡ ✐ ♦ ♥ ❛ ❧ ❡ ✉ ❡ ❝ ✉ ❛ ♥ ❧ ❛ ▼ ❛ ❡ ❛ ❡ ♥ ■ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ❱ ✐ ❛ ❧ ✱ ❡ ❡ ❡ ♠ ❛ ♥ ♦ ❡  

✐ ♥ ❝ ❧ ✉ ② ❡ ❡ ♥ ✉ ♠ ❛ ❧ ❧ ❛ ❝ ✉ ✐ ❝ ✉ ❧ ❛ ✳

❈ ♦ ♠ ❜ ✐ ♥ ❛ ♥ ❞ ♦ ❡ ♥ ♦ ♥ ❝ ❡ ✉ ♥ ♣ ♦ ♣ ✐ ♦ ❝ ❧ ❛ ❛ ♠ ❡ ♥ ❡ ❞ ❡ ✜ ♥ ✐ ❞ ♦ ❝ ♦ ♥ ✉ ♥ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❞ ❡ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❞ ✐ ✲  

♣ ♦ ♥ ✐ ❜ ❧ ❡ ② ❛ ♣ ❛ ♣ ❛ ❛ ❛ ❧ ✜ ♥ ✱ ✉ ❣ ❡ ❡ ❧ ❝ ✉ ♦ ✉ ❡ ♠ ❛ ♣ ♦ ♣ ✐ ❛ ❞ ❛ ♠ ❡ ♥ ❡ ❡ ❞ ❡ ♥ ♦ ♠ ✐ ♥ ❛ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥  

♣ ❛ ❛ ■ ♥ ❣ ❡ ♥ ✐ ❡ ♦ ❝ ♦ ♥ ❍ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ■ ♥ ❢ ♦ ♠ ✐ ❝ ❛ ② ❡ ❛ ❝ ♦ ♠ ♣ ✐ ❧ ❛ ❝ ✐ ♥ ❞ ❡ ❛ ♣ ✉ ♥ ❡ ✉ ❡ ✐ ❡ ♥ ❡ ♥ ❝ ♦ ♠ ♦  

♣ ♦ ♣ ✐ ♦ ❡ ✈ ✐ ❝ ♦ ♠ ♦ ❣ ✉ ❛ ② ❡ ❢ ❡ ❡ ♥ ❝ ✐ ❛ ♣ ❛ ❛ ❧ ♦ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ✱ ❞ ❡ ❜ ✐ ❞ ♦ ❛ ❧ ❛ ♣ ♦ ❝ ❛ ❞ ✐ ♣ ♦ ♥ ✐ ❜ ✐ ❧ ✐ ❞ ❛ ❞  

❞ ❡ ♠ ❛ ❡ ✐ ❛ ❧ ♦ ❜ ❡ ❡ ❧ ❡ ♠ ❛ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ✳ ❈ ❛ ❜ ❡ ❛ ❝ ❧ ❛ ❛ ✉ ❡ ❡ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ❧ ❝ ✉ ♦ ② ♣ ♦ ❡ ♥ ❞ ❡ ❞ ❡ ❡ ♦  

❛ ♣ ✉ ♥ ❡ ✱ ❡ ♣ ❡ ❡ ♥ ❛ ❛ ❧ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ② ❛ ❧ ♣ ♦ ❢ ❡ ✐ ♦ ♥ ❛ ❧ ✉ ♥ ❡ ♥ ❢ ♦ ✉ ❡ ♣ ❝ ✐ ❝ ♦ ② ❛ ♣ ❧ ✐ ❝ ❛ ❞ ♦ ❞ ❡ ❧ ❛ ♠ ❡ ♦ ❞ ♦ ✲  

❧ ♦ ❣ ❛ ❞ ✐ ❝ ✉ ✐ ❞ ❛ ✱ ❡ ♥ ✈ ❡ ③ ❞ ❡ ♣ ♦ ❢ ✉ ♥ ❞ ✐ ③ ❛ ❡ ♥ ❡ ❧ ❛ ♣ ❡ ❝ ♦ ❡ ✐ ❝ ♦ ② ❡ ❧ ✐ ❣ ♦ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ❞ ❡ ❧ ♦ ❡ ♠ ❛  

② ❛ ✉ ❡ ♦ ❜ ❡ ❡ ♦ ❡ ① ✐ ❡ ✉ ✜ ❝ ✐ ❡ ♥ ❡ ♠ ❛ ❡ ✐ ❛ ❧ ❞ ❡ ♠ ✉ ② ❜ ✉ ❡ ♥ ❛ ❝ ❛ ❧ ✐ ❞ ❛ ❞ ② ✉ ❡ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ❞ ✐ ♣ ♦ ♥ ✐ ❜ ❧ ❡ ✱  

♣ ♦ ❧ ♦ ✉ ❡ ❡ ♣ ✐ ❞ ❡ ❛ ❧ ♦ ❧ ❡ ❝ ♦ ❡ ♠ ❛ ♥ ❡ ♥ ❡ ❡ ♦ ❡ ♥ ♠ ❡ ♥ ❡ ❛ ❧ ✉ ✐ ❧ ✐ ③ ❛ ❧ ♦ ❛ ♣ ✉ ♥ ❡ ✳ ❊ ❛ ♥ ♦ ❛ ♥ ♦  

♣ ❡ ❡ ♥ ❞ ❡ ♥ ❡ ✈ ✐ ❝ ♦ ♠ ♦ ❧ ✐ ❜ ♦ ❞ ❡ ❡ ① ♦ ✱ ♣ ♦ ❧ ♦ ✉ ❡ ✐ ❡ ❡ ✉ ✐ ❡ ❡ ♣ ♦ ❢ ✉ ♥ ❞ ✐ ③ ❛ ❡ ♥ ❧ ❛ ♣ ❛ ❡ ❡ ✐ ❝ ❛ ②  

❝ ♦ ♥ ❝ ❡ ♣ ✉ ❛ ❧ ❞ ❡ ❧ ♦ ♠ ♦ ❞ ♦ ② ❧ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❛ ✉ ♣ ❡ ❡ ♥ ❛ ❞ ❛ ✱ ❡ ♥ ❧ ❛ ❡ ❝ ❝ ✐ ♥ ❞ ❡ ❇ ✐ ❜ ❧ ✐ ♦ ❣ ❛ ❢ ❛ ❡  

❡ ♥ ❝ ✉ ❡ ♥ ❛ ✉ ♥ ❛ ❧ ✐ ❛ ❞ ❡ ❡ ❢ ❡ ❡ ♥ ❝ ✐ ❛ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ♥ ❡ ❝ ♦ ♥ ✉ ❧ ❛ ❞ ❛ ✳

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✷ ✳ ❈ ❖ ◆ ❈ ❊ ❚ ❖ ❙ ❇ ➪ ❙ ■ ❈ ❖ ❙ ❉ ❊ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆  

❋ ✐ ❣ ✉ ❛ ✷ ✳ ✶ ✳ ✿ ♦ ❝ ❡ ♦ ❞ ❡ ❙ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥  

❼ 

❈ ♦ ♥ ✐ ♥ ✉ ❛ ✿   ▲ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ♣ ❡ ❡ ♥ ❡ ❝ ❡ ♥ ❛ ❧ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❞ ❡ ❧ ♦ ♥ ♠ ❡ ♦ ❡ ❛ ❧ ❡ ②  

♣ ✉ ❡ ❞ ❡ ♥ ♦ ♠ ❛ ❝ ✉ ❛ ❧ ✉ ✐ ❡ ✈ ❛ ❧ ♦ ❡ ♥ ❞ ✐ ❝ ❤ ♦ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ✳

❼ 

❉ ✐ ❝ ❡ ❛ ✿   ▲ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ♦ ❧ ♦ ♣ ✉ ❡ ❞ ❡ ♥ ♦ ♠ ❛ ❝ ✐ ❡ ♦ ✈ ❛ ❧ ♦ ❡ ✳ ❯ ♥ ❝ ❛ ♦ ❡ ♣ ❡ ❝ ✐ ❛ ❧ ❡ ❡ ❧ ❝ ❛ ♦  

❞ ❡ ❧ ❛ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ❡ ♥ ❡ ❛ ✳

❼ 

❈ ♦ ♠ ❜ ✐ ♥ ❛ ♦ ✐ ❛ ✿   ▲ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ♦ ❧ ♦ ♣ ✉ ❡ ❞ ❡ ♥ ♦ ♠ ❛ ✈ ❛ ❧ ♦ ❡ ❞ ❡ ♥ ♦ ❞ ❡ ✉ ♥ ❝ ♦ ♥ ✲  

 ❥ ✉ ♥ ♦ ❞ ❡ ✜ ♥ ✐ ❞ ♦ ② ❡ ❜ ✉ ❝ ❛ ❡ ♥ ❝ ♦ ♥ ❛ ❧ ❛ ❝ ♦ ♠ ❜ ✐ ♥ ❛ ❝ ✐ ♥ ❞ ❡ ✈ ❛ ❧ ♦ ❡ ♣ ❛ ❛ ❧ ❛ ❞ ✐ ✐ ♥ ❛ ✈ ❛ ✐ ❛ ✲ 

❜ ❧ ❡ ✉ ❡ ❤ ❛ ❝ ❡ ♣ ✐ ♠ ♦ ❡ ❧ ❡ ✉ ❧ ❛ ❞ ♦ ✳

♦ ❧ ❛ ❝ ❛ ♥ ✐ ❞ ❛ ❞ ❞ ❡ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✿

❼ 

❯ ♥ ✐ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✿   ❯ ♥ ♦ ❧ ♦ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ♦ ❢ ✉ ♥ ❝ ✐ ♥ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✳

❼ 

▼ ✉ ❧ ✐ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✿   ❉ ♦ ♦ ♠ ❛ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✉ ❡ ❝ ♦ ♠ ♣ ✐ ❡ ♥ ❡ ♥ ❡ ♣ ♦ ❧ ♦ ❡ ❝ ✉ ♦ ✳

❊ ♦ ♦ ♥ ♦ ❧ ❛ ♠ ❡ ♥ ❡ ❛ ❧ ❣ ✉ ♥ ♦ ❞ ❡ ❧ ♦ ❝ ✐ ❡ ✐ ♦ ❞ ❡ ❝ ❧ ❛ ✐ ✜ ❝ ❛ ❝ ✐ ♥ ❞ ❡ ✉ ♥ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✱  

♣ ❡ ♦ ❝ ❛ ❞ ❛ ✉ ♥ ♦ ❞ ❡ ❡ ❧ ❧ ♦ ♣ ✉ ❡ ❞ ❡ ❛ ✉ ✈ ❡ ③ ❞ ✐ ✈ ✐ ❞ ✐ ❡ ❡ ♥ ♦ ❛ ❝ ❛ ❡ ❣ ♦ ❛ ✳

❊ ♥ ❡ ❧ ❡ ♥ ✐ ❞ ♦ ❡ ❝ ❛ ♠ ❡ ♥ ❡ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ✱ ✉ ♥ ❛ ♦ ❧ ✉ ❝ ✐ ♥ ♣ ✐ ♠ ❛ ❡ ❛ ♦ ❝ ✐ ❛ ❞ ❛ ❞ ✐ ❡ ❝ ❛ ♠ ❡ ♥ ❡ ❝ ♦ ♥  

✉ ♥ ♣ ✉ ♥ ♦ ❞ ❡ ✐ ♥ ✢ ❡ ① ✐ ♥ ❞ ❡ ✉ ♥ ❛ ❢ ✉ ♥ ❝ ✐ ♥ ✱ ❡ ❧ ❝ ✉ ❛ ❧ ✱ ❞ ❡ ♣ ❡ ♥ ❞ ✐ ❡ ♥ ❞ ♦ ❞ ❡ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✱ ♣ ✉ ❡ ❞ ❡ ❛ ❛ ❡ ❞ ❡ ✉ ♥  

♠ ① ✐ ♠ ♦  

♦ ✉ ♥ ♠ ♥ ✐ ♠ ♦  

✱ ♣ ♦ ❧ ♦ ✉ ❡ ❝ ✉ ❛ ♥ ❞ ♦ ❤ ❛ ❜ ❧ ❛ ♠ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ✱ ❡ ❛ ♠ ♦ ❤ ❛ ❜ ❧ ❛ ♥ ❞ ♦ ❞ ❡ ❡ ♥ ❝ ♦ ♥ ❛  

♣ ❛ ❛ ✉ ♥ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❞ ❡ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ✱ ❧ ♦ ✈ ❛ ❧ ♦ ❡ ✉ ❡ ♠ ❛ ① ✐ ♠ ✐ ③ ❛ ♥ ♦ ♠ ✐ ♥ ✐ ♠ ✐ ③ ❛ ♥ ✉ ♥ ❛ ❢ ✉ ♥ ❝ ✐ ♥  

♦ ❜ ❥ ❡ ✐ ✈ ♦ ❛ ❧ ♠ ✐ ♠ ♦ ✐ ❡ ♠ ♣ ♦ ✉ ❡ ❛ ✐ ❢ ❛ ❝ ❡ ♥ ✉ ♥ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❞ ❡ ❡ ✐ ❝ ❝ ✐ ♦ ♥ ❡ ✉ ❡ ❡ ① ✐ ❡ ♥ ❡ ♥ ❡ ❧ ♠ ♦ ❞ ❡ ❧ ♦ ✳

✷ ✳ ✷ ✳ ❉ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ✉ ♥ ▼ ♦ ❞ ❡ ❧ ♦ ♣ ❛ ❛ ❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥  

❆ ❝ ♦ ♥ ✐ ♥ ✉ ❛ ❝ ✐ ♥ ❡ ❞ ❡ ❝ ✐ ❜ ❡ ✉ ♥ ♣ ♦ ❝ ❡ ❞ ✐ ♠ ✐ ❡ ♥ ♦ ♣ ❛ ❛ ❡ ❧ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ✉ ♥ ♠ ♦ ❞ ❡ ❧ ♦ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ✳

❆ ❧ ♣ ✐ ♥ ❝ ✐ ♣ ✐ ♦ ✱ ♠ ✐ ❡ ♥ ❛ ❡ ❧ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ❡ ❛ ❝ ♦ ✉ ♠ ❜ ❛ ❛ ❡ ❛ ❧ ✐ ③ ❛ ❡ ❧ ♣ ♦ ❝ ❡ ♦ ❞ ❡ ♣ ❡ ♥ ❛ ♠ ✐ ❡ ♥ ♦ ♣ ❛ ❛  

♠ ♦ ❞ ❡ ❧ ❛ ✱ ❡ ✉ ❣ ✐ ❡ ❡ ❡ ❣ ✉ ✐ ♣ ❛ ♦ ❛ ♣ ❛ ♦ ❡ ❡ ❛ ❧ ❣ ♦ ✐ ♠ ♦ ❡ ♥ ❢ ♦ ♠ ❛ ✐ ❡ ♠ ✐ ❝ ❛ ✳ ❈ ♦ ♥ ❡ ❧ ♣ ❛ ♦ ❞ ❡ ❧  

✐ ❡ ♠ ♣ ♦ ② ❧ ❛ ♣ ❝ ✐ ❝ ❛ ❝ ♦ ♥ ❛ ♥ ❡ ✱ ❡ ❧ ♣ ♦ ❝ ❡ ♦ ❡ ✈ ♦ ❧ ✈ ❡ ❛ ❧ ❣ ♦ ♥ ❛ ✉ ❛ ❧ ❡ ✐ ♥ ✉ ✐ ✐ ✈ ♦ ✳

✶ ✳ ❊ ♥ ❡ ♥ ❞ ❡ ❡ ❧ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ✿   ❆ ✉ ♥ ✉ ❡ ♣ ✉ ❡ ❞ ❛ ♣ ❛ ❡ ❝ ❡ ♦ ❜ ✈ ✐ ♦ ❡ ✐ ♥ ♥ ❡ ❝ ❡ ❛ ✐ ♦ ✐ ♥ ❝ ❧ ✉ ❡ ❡ ♣ ❛ ♦ ❡ ① ♣ ❧ ✲  

❝ ✐ ❛ ♠ ❡ ♥ ❡ ✱ ❧ ❛ ❡ ① ♣ ❡ ✐ ❡ ♥ ❝ ✐ ❛ ❡ ♥ ❧ ❛ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ♠ ✉ ❡ ❛ ✉ ❡ ❧ ❛ ❝ ❛ ✉ ❛ ♣ ✐ ♥ ❝ ✐ ♣ ❛ ❧ ♣ ♦  

❧ ❛ ✉ ❡ ❡ ❢ ❛ ❝ ❛ ❛ ✱ ❡ ♣ ❡ ❝ ✐ ❛ ♠ ❡ ♥ ❡ ♣ ♦ ✉ ❡ ♥ ♦ ❡ ❤ ❛ ❞ ❡ ✜ ♥ ✐ ❞ ♦ ② ❞ ❡ ❧ ✐ ♠ ✐ ❛ ❞ ♦ ❝ ♦ ❡ ❝ ❛ ♠ ❡ ♥ ❡  

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10/86

✷ ✳ ❈ ❖ ◆ ❈ ❊ ❚ ❖ ❙ ❇ ➪ ❙ ■ ❈ ❖ ❙ ❉ ❊ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆  

❡ ❧ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❛ ❡ ♦ ❧ ✈ ❡ ♣ ♦ ❧ ♦ ✉ ❡ ♥ ♦ ❡ ① ✐ ❡ ✉ ♥ ❛ ❝ ♦ ♠ ♣ ❡ ♥ ✐ ♥ ❞ ❡ ❧ ♠ ✐ ♠ ♦ ✳ ❛ ❛ ❛ ❡ ❣ ✉ ❛ ❡ ❧  

❝ ♦ ❡ ❝ ♦ ❡ ♥ ❡ ♥ ❞ ✐ ♠ ✐ ❡ ♥ ♦ ❞ ❡ ❧ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ✱ ❤ ❛ ② ✈ ❛ ✐ ❛ ❡ ❛ ❡ ❣ ✐ ❛ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ♥ ❡ ❣ ✉ ✐ ❡ ✱ ❡ ♥ ❡ ❧ ❛  

❝ ✉ ❛ ❧ ❡ ✉ ♥ ❛ ❞ ❡ ❧ ❛ ♠ ✐ ❧ ❡ ❡ ❤ ❛ ❝ ❡ ✉ ♥ ❡ ✉ ❡ ♠ ❛ ✱ ❞ ✐ ❜ ✉ ❥ ♦ ♦ ❞ ✐ ❛ ❣ ❛ ♠ ❛ ❞ ❡ ❧ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ✱ ❞ ❡ ✉ ♥ ❛  

❢ ♦ ♠ ❛ ✐ ♠ ♣ ❧ ❡ ♣ ❡ ♦ ❛ ❛ ♥ ❞ ♦ ❞ ❡ ❡ ♣ ❡ ❡ ♥ ❛ ❣ ✜ ❝ ❛ ♠ ❡ ♥ ❡ ♦ ❞ ♦ ❧ ♦ ❛ ♣ ❡ ❝ ♦ ❞ ❡ ❧ ❛ ✐ ✉ ❛ ❝ ✐ ♥ ✳

✷ ✳ ■ ❞ ❡ ♥ ✐ ✜ ❝ ❛ ❡ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✿   ❈ ♦ ♠ ♦ ❡ ♠ ❡ ♥ ❝ ✐ ♦ ♥ ❡ ♥ ❧ ❛ ❡ ❝ ❝ ✐ ♥ ❛ ♥ ❡ ✐ ♦ ✱ ✉ ♥ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ✲  

❝ ✐ ♥ ❜ ✉ ❝ ❛ ✐ ❡ ♠ ♣ ❡ ✉ ♥ ♦ ❞ ❡ ❞ ♦ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ♣ ♦ ✐ ❜ ❧ ❡ ✱ ♠ ❛ ① ✐ ♠ ✐ ③ ❛ ♦ ♠ ✐ ♥ ✐ ♠ ✐ ③ ❛ ❛ ❧ ❣ ♦ ✳ ❊ ♥ ❢ ✉ ♥ ❝ ✐ ♥ ❛  

❧ ❛ ✐ ♥ ❢ ♦ ♠ ❛ ❝ ✐ ♥ ♣ ♦ ♣ ♦ ❝ ✐ ♦ ♥ ❛ ❞ ❛ ② ❛ ❧ ❝ ♦ ♥ ❡ ① ♦ ❞ ❡ ❧ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❡ ♥ ❝ ✉ ❡ ✐ ♥ ✱ ❡ ♣ ♦ ✐ ❜ ❧ ❡ ✐ ❞ ❡ ♥ ✐ ✜ ❝ ❛  

❞ ✐ ❝ ❤ ♦ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✳ ♦ ❡ ❥ ❡ ♠ ♣ ❧ ♦ ✱ ✐ ❡ ♥ ✉ ♥ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❡ ❡ ❛ ❛ ♥ ❞ ♦ ❝ ♦ ♥ ❛ ♣ ❡ ❝ ♦ ❝ ♦ ♠ ♦ ❝ ♦ ♦ ✱  

❞ ✐ ❛ ♥ ❝ ✐ ❛ ✱ ♠ ❛ ♥ ♦ ❞ ❡ ♦ ❜ ❛ ♦ ♠ ❛ ❡ ✐ ❛ ❧ ❡ ✱ ❡ ❢ ❝ ✐ ❧ ❧ ❧ ❡ ❣ ❛ ❛ ❧ ❛ ❝ ♦ ♥ ❝ ❧ ✉ ✐ ♥ ✉ ❡ ❧ ♦ ✉ ❡ ❡ ❜ ✉ ❝ ❛  

❡ ♠ ✐ ♥ ✐ ♠ ✐ ③ ❛ ❞ ✐ ❝ ❤ ♦ ❛ ♣ ❡ ❝ ♦ ✳ ♦ ❡ ❧ ❝ ♦ ♥ ❛ ✐ ♦ ✱ ✐ ❡ ❡ ❤ ❛ ❜ ❧ ❛ ♥ ❞ ♦ ❞ ❡ ❝ ♦ ❛ ❝ ♦ ♠ ♦ ✈ ❡ ♥ ❛ ✱  

✐ ♥ ❣ ❡ ♦ ✱ ❞ ✐ ♥ ❡ ♦ ✱ ❣ ❛ ♥ ❛ ♥ ❝ ✐ ❛ ✱ ❡ ♥ ❞ ✐ ♠ ✐ ❡ ♥ ♦ ✱ ❡ ❝ ✳ ♣ ✉ ❡ ❞ ❡ ✐ ♥ ❢ ❡ ✐ ❡ ✉ ❡ ❡ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ♣ ❡ ❡ ❣ ✉ ✐ ❞ ♦ ❡  

♠ ❛ ① ✐ ♠ ✐ ③ ❛ ❛ ❧ ❡ ❝ ♦ ❛ ✳

✸ ✳ ■ ❞ ❡ ♥ ✐ ✜ ❝ ❛ ❧ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ✿   ▲ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ♦ ♥ ✱ ❝ ♦ ♠ ♦ ✉ ♥ ♦ ♠ ❜ ❡ ❧ ♦  

✉ ❣ ✐ ❡ ❡ ✱ ❛ ✉ ❡ ❧ ❧ ♦ ❛ ♣ ❡ ❝ ♦ ❞ ❡ ❧ ♠ ♦ ❞ ❡ ❧ ♦ ✉ ❡ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ♥ ❜ ❛ ❥ ♦ ❡ ❧ ❝ ♦ ♥ ♦ ❧ ❞ ❡ ❧ ♦ ♠ ❛ ❞ ♦ ❞ ❡  

❞ ❡ ❝ ✐ ✐ ♦ ♥ ❡ ✱ ❞ ❡ ♠ ❛ ♥ ❡ ❛ ✉ ❡ ❛ ❥ ✉ ❛ ♥ ❞ ♦ ❡ ❧ ✈ ❛ ❧ ♦ ❞ ❡ ❞ ✐ ❝ ❤ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✱ ❡ ♠ ♦ ❞ ✐ ✜ ❝ ❛ ❡ ❧ ❡ ✉ ❧ ❛ ❞ ♦  

❞ ❡ ❧ ♠ ♦ ❞ ❡ ❧ ♦ ✳ ● ❡ ♥ ❡ ❛ ❧ ♠ ❡ ♥ ❡ ❡ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ♦ ♥ ❡ ♣ ❡ ❡ ♥ ❛ ❞ ❛ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ♠ ❡ ♥ ❡ ❡ ♥ ❢ ♦ ♠ ❛ ❞ ❡  

✉ ♥ ✈ ❡ ❝ ♦ ❞ ❡ ❧ ❛ ❢ ♦ ♠ ❛  

X   =   x1, x2, x3,... ❡ ❣ ♥ ❡ ❛ ❧ ❛ ❝ ❛ ♥ ✐ ❞ ❛ ❞ ❞ ❡ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✉ ❡ ❡ ① ✐ ❡ ♥ ❡ ♥  ❡ ❧ ♠ ♦ ❞ ❡ ❧ ♦ ✳

✹ ✳ ❊ ① ♣ ❡ ❛ ❡ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❝ ♦ ♠ ♦ ✉ ♥ ❛ ❢ ✉ ♥ ❝ ✐ ♥ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ❞ ❡ ❧ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ✿   ❊ ❡ ♣ ❛ ♦  

❝ ♦ ♥ ✐ ❡ ❡ ♥ ✐ ❞ ❡ ♥ ✐ ✜ ❝ ❛ ❧ ❛ ❡ ❧ ❛ ❝ ✐ ♦ ♥ ❡ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ❡ ① ✐ ❡ ♥ ❡ ❡ ♥ ❡ ❧ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✲  

✐ ♥ ✉ ❡ ♣ ♦ ❞ ✉ ❝ ❡ ♥ ❡ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ❧ ♣ ❛ ♦ ✷ ✳ ❆ ❡ ❛ ❢ ✉ ♥ ❝ ✐ ♥ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ❡ ❧ ❡ ❧ ❧ ❛ ♠ ❛ ❋ ✉ ♥ ❝ ✐ ♥  

❖ ❜ ❥ ❡ ✐ ✈ ♦ ✳ ❊ ① ♣ ❡ ❛ ❞ ♦ ❡ ♥ ❢ ♦ ♠ ❛ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ✿

max ó min Z  = f (x1, x2, x3,...)

✺ ✳ ■ ❞ ❡ ♥ ✐ ✜ ❝ ❛ ❧ ❛ ❡ ✐ ❝ ❝ ✐ ♦ ♥ ❡ ❡ ♥ ❡ ❧ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ✿   ❯ ♥ ❛ ❡ ✐ ❝ ❝ ✐ ♥ ❡ ♥ ✉ ♥ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ✲  

❝ ✐ ♥ ❡ ❡ ✜ ❡ ❡ ❛ ❝ ✉ ❛ ❧ ✉ ✐ ❡ ♦ ❜ ❝ ✉ ❧ ♦ ✉ ❡ ♥ ♦ ✐ ♠ ♣ ✐ ❞ ❡ ♦ ❜ ❡ ♥ ❡ ✉ ♥ ❡ ✉ ❧ ❛ ❞ ♦ ✐ ❞ ❡ ❛ ❧ ♦ ❞ ❡ ❡ ❛ ❞ ♦ ✳

● ❡ ♥ ❡ ❛ ❧ ♠ ❡ ♥ ❡ ❧ ❛ ❡ ✐ ❝ ❝ ✐ ♦ ♥ ❡ ❡ ♦ ✐ ❣ ✐ ♥ ❛ ♥ ❞ ❡ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✐ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ✿

❛  ✮ ▲ ✐ ♠ ✐ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ❡ ❝ ✉ ♦ ❞ ✐ ♣ ♦ ♥ ✐ ❜ ❧ ❡ ✱ ❛ ❧ ❡ ❝ ♦ ♠ ♦ ♠ ❛ ❡ ✐ ❛ ♣ ✐ ♠ ❛ ✱ ✐ ❡ ♠ ♣ ♦ ✱ ❡ ♣ ❛ ❝ ✐ ♦ ✱ ♣ ❡ ✲  

♦ ♥ ❛ ❧ ✱ ❡ ❝ ✳

❜  ✮ ❘ ❡ ✉ ❡ ✐ ♠ ✐ ❡ ♥ ♦ ✉ ❡ ❞ ❡ ❜ ❡ ♥ ❡ ❝ ✉ ♠ ♣ ❧ ✐ ❞ ♦ ✱ ❛ ❧ ❡ ❝ ♦ ♠ ♦ ❝ ❛ ♥ ✐ ❞ ❛ ❞ ❡ ♠ ♥ ✐ ♠ ❛ ❞ ❡ ♣ ♦ ❞ ✉ ❝ ✲  

❝ ✐ ♥ ✱ ❡ ♣ ❡ ❝ ✐ ✜ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ♣ ♦ ❞ ✉ ❝ ♦ ✱ ❝ ❛ ❧ ✐ ❞ ❛ ❞ ✱ ❡ ❝ ✳

❝  ✮ ❈ ♦ ♥ ❞ ✐ ❝ ✐ ♦ ♥ ❡ ♦ ❜ ❡ ❧ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ♣ ❛ ❛ ✉ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ❡ ♥ ❧ ❛ ♣ ❝ ✐ ❝ ❛ ✱ ❝ ♦ ♠ ♦ ♣ ♦  

❡ ❥ ❡ ♠ ♣ ❧ ♦ ✱ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ♥ ♦ ♥ ❡ ❣ ❛ ✐ ✈ ❛ ✱ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❡ ♥ ❡ ❛ ✱ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❜ ✐ ♥ ❛ ✐ ❛ ✱ ❡ ❝ ✳

✻ ✳ ❊ ① ♣ ❡ ❛ ❧ ❛ ❡ ✐ ❝ ❝ ✐ ♥ ❡ ❝ ♦ ♠ ♦ ❡ ❧ ❛ ❝ ✐ ♦ ♥ ❡ ❡ ♥ ❡ ❧ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ❞ ❡ ❞ ❡ ❝ ✐ ✐ ♥ ✿ ▲ ❛ ❡ ❧ ❛ ❝ ✐ ♦ ♥ ❡  

♣ ❛ ❛ ❡ ♣ ❡ ❡ ♥ ❛ ❧ ❛ ❡ ✐ ❝ ❝ ✐ ♦ ♥ ❡ ♥ ♦ ❡ ❧ ✐ ♠ ✐ ❛ ♥ ♦ ❧ ❛ ♠ ❡ ♥ ❡ ❛ ❡ ❧ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ✐ ❣ ✉ ❛ ❧ ❞ ❛ ❞ ✐ ♥ ♦ ✉ ❡  

♣ ✉ ❡ ❞ ❡ ♥ ❡ ① ♣ ❡ ❛ ❡ ❛ ♠ ❜ ✐ ♥ ❝ ♦ ♠ ♦ ❞ ❡ ✐ ❣ ✉ ❛ ❧ ❞ ❛ ❞ ❡ ✱ ♣ ♦ ❧ ♦ ✉ ❡ ❡ ✐ ♠ ♣ ♦ ❛ ♥ ❡ ❡ ♥ ❡ ♠ ✉ ② ❝ ❧ ❛ ❛  

❧ ❛ ♥ ❛ ✉ ❛ ❧ ❡ ③ ❛ ❞ ❡ ❧ ❛ ❡ ✐ ❝ ❝ ✐ ♥ ✱ ❡ ❞ ❡ ❝ ✐ ✱ ❛ ❜ ❡ ✐ ✿

❛  ✮  ❉ ❡ ❜ ❡ ❝ ✉ ♠ ♣ ❧ ✐ ❡ ❝ ♦ ♥ ✉ ♥ ✈ ❛ ❧ ♦ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ✿   ❘ ❡ ✐ ❝ ❝ ✐ ♥ ❞ ❡ ✐ ❣ ✉ ❛ ❧ ❞ ❛ ❞  =

❜  ✮  ❊ ① ✐ ❡ ✉ ♥ ✈ ❛ ❧ ♦ ♠ ♥ ✐ ♠ ♦ ✉ ❡ ❞ ❡ ❜ ❡ ❛ ❧ ❝ ❛ ♥ ③ ❛ ❡ ✿   ❘ ❡ ✐ ❝ ❝ ✐ ♥ ❞ ❡ ❞ ❡ ✐ ❣ ✉ ❛ ❧ ❞ ❛ ❞ ❞ ❡ ❧ ✐ ♣ ♦  ≥

❝  ✮  ❊ ① ✐ ❡ ✉ ♥ ✈ ❛ ❧ ♦ ♠ ① ✐ ♠ ♦ ✉ ❡ ♥ ♦ ♣ ✉ ❡ ❞ ❡ ❡ ① ❝ ❡ ❞ ❡ ❡ ✿   ❘ ❡ ✐ ❝ ❝ ✐ ♥ ❞ ❡ ❞ ❡ ✐ ❣ ✉ ❛ ❧ ❞ ❛ ❞ ❞ ❡ ❧ ✐ ♣ ♦  

≤

✼ ✳ ❊ ① ♣ ❡ ❛ ❡ ❧ ♠ ♦ ❞ ❡ ❧ ♦ ❡ ♥ ✉ ♥ ❛ ❢ ♦ ♠ ❛ ❝ ❛ ♥ ♥ ✐ ❝ ❛ ♦ ❡ ♥ ❞ ❛ ✿ ❊ ♦ ♣ ❡ ♠ ✐ ✐ ✐ ❞ ❡ ♥ ✐ ✜ ❝ ❛ ② ❛ ♣ ❧ ✐ ❝ ❛  

❝ ♦ ❡ ❝ ❛ ♠ ❡ ♥ ❡ ❡ ❧ ❛ ❧ ❣ ♦ ✐ ♠ ♦ ❞ ❡ ♦ ❧ ✉ ❝ ✐ ♥ ❡ ♥ ❧ ❛ ❡ ❛ ♣ ❛ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ❡ ♥ ❡ ✳ ❈ ✐ ❡ ♦ ❛ ❧ ❣ ♦ ✐ ♠ ♦  

❡ ✉ ✐ ❡ ❡ ♥ ✉ ❡ ❡ ❧ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❡ ❡ ① ♣ ❡ ❛ ❞ ♦ ❡ ♥ ✉ ♥ ❛ ❢ ♦ ♠ ❛ ❞ ❡ ✜ ♥ ✐ ❞ ❛ ♣ ❛ ❛ ✐ ♥ ✐ ❝ ✐ ❛ ❡ ❧ ♣ ♦ ❝ ❡ ♦ ❞ ❡  

♦ ❧ ✉ ❝ ✐ ♥ ✳

❊ ❡ ♣ ♦ ❝ ❡ ♦ ❡ ❡ ♥ ❡ ♥ ❞ ❡ ♠ ❛ ❝ ❧ ❛ ❛ ♠ ❡ ♥ ❡ ❡ ♥ ❧ ♦ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❝ ❛ ♣ ✉ ❧ ♦ ❝ ✉ ❛ ♥ ❞ ♦ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ❡  

❡ ♥ ❡ ❧ ❝ ♦ ♥ ❡ ① ♦ ❞ ❡ ✉ ♥ ❝ ❛ ♦ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ✳

✶ ✵ 

8/19/2019 Notas Optimizacion Ingenieros-libre

11/86

✷ ✳ ❈ ❖ ◆ ❈ ❊ ❚ ❖ ❙ ❇ ➪ ❙ ■ ❈ ❖ ❙ ❉ ❊ ❖ ❚ ■ ▼ ■ ❩ ❆ ❈ ■ ◆  

❋ ✐ ❣ ✉ ❛ ✷ ✳ ✷ ✳ ✿ ♦ ❝ ❡ ♦ ♣ ❛ ❛ ❡ ❧ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ✉ ♥ ♠ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥  

✶ ✶ 

8/19/2019 Notas Optimizacion Ingenieros-libre

12/86

✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙      

❈ ♦ ♠ ♦ ❡ ♠ ❡ ♥ ❝ ✐ ♦ ♥ ❡ ♥ ❧ ❛ ✐ ♥ ♦ ❞ ✉ ❝ ❝ ✐ ♥ ✱ ❡ ❡ ❡ ✉ ♥ ❝ ✉ ♦ ❡ ♠ ✐ ♥ ❡ ♥ ❡ ♠ ❡ ♥ ❡ ♣ ❝ ✐ ❝ ♦ ② ❛ ♣ ❧ ✐ ❝ ❛ ❞ ♦ ✱ ♣ ♦  

❧ ♦ ✉ ❡ ❧ ❛ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ❧ ♦ ♠ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ❡ ❤ ❛ ♥ ✐ ❝ ❛ ♠ ❡ ♥ ❡ ♣ ♦ ♠ ❡ ❞ ✐ ♦ ❞ ❡ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛  

✐ ♥ ❢ ♦ ♠ ✐ ❝ ❛ ✱ ❛ ✉ ❡ ❡ ♥ ❡ ❡ ❝ ❛ ♣ ✉ ❧ ♦ ❡ ❤ ❛ ✉ ♥ ❛ ❜ ❡ ✈ ❡ ✐ ♥ ♦ ❞ ✉ ❝ ❝ ✐ ♥ ❛ ❧ ❛ ♠ ✐ ♠ ❛ ✳ ❙ ❡ ♣ ❡ ❡ ♥ ❛ ♥  

❡ ♥ ❧ ❛ ❡ ❝ ❝ ✐ ♦ ♥ ❞ ❡ ❜ ✐ ❜ ❧ ✐ ♦ ❣ ❛ ❢ ❛ ✈ ❛ ✐ ❛ ❡ ❢ ❡ ❡ ♥ ❝ ✐ ❛ ❞ ♦ ♥ ❞ ❡ ❡ ❧ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ♣ ✉ ❡ ❞ ❡ ❝ ♦ ♥ ✉ ❧ ❛ ❡ ♥ ❞ ❡ ❛ ❧ ❧ ❡ ❧ ❛  

❡ ♦ ❛ ❞ ❡ ❞ ❡ ❝ ❛ ❞ ❛ ✉ ♥ ♦ ❞ ❡ ❧ ♦ ❛ ❧ ❣ ♦ ✐ ♠ ♦ ✱ ❛ ❝ ♦ ♠ ♦ ✉ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ❡ ♥ ❢ ♦ ♠ ❛ ♠ ❛ ♥ ✉ ❛ ❧ ✐ ♥ ❡ ❝ ❡ ✐ ❛  

❝ ♦ ♥ ♦ ❝ ❡ ❧ ♦ ✳

✸ ✳ ✶ ✳ ■ ♥ ♦ ❞ ✉ ❝ ❝ ✐ ♥ ❛ ❧ ❙ ♦ ❢ ✇ ❛ ❡ ▲ ✐ ❜ ❡ ❞ ❡ ❈ ❞ ✐ ❣ ♦ ❆ ❜ ✐ ❡ ♦  

❊ ❧ ♠ ♦ ✈ ✐ ♠ ✐ ❡ ♥ ♦ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ❞ ❡ ❝ ❞ ✐ ❣ ♦ ❛ ❜ ✐ ❡ ♦ ✭ ❋ ❡ ❡ ❛ ♥ ❞ ❖ ♣ ❡ ♥ ❙ ♦ ✉ ❝ ❡ ❙ ♦ ❢ ✇ ❛ ❡ ♦ ❋ ❖ ❙ ❙ ✮  

✶ 

✉ ❣ ✐ ❞ ✉ ❛ ♥ ❡ ❧ ❛ ❞ ❝ ❛ ❞ ❛ ❞ ❡ ✶ ✾ ✽ ✵ ❝ ♦ ♠ ♦ ✉ ♥ ❛ ❡ ♣ ✉ ❡ ❛ ❛ ❧ ❝ ❛ ♠ ❜ ✐ ♦ ❡ ♥ ❡ ❧ ♠ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ♥ ❡ ❣ ♦ ❝ ✐ ♦ ❡ ♥ ❡ ❧  

♠ ✉ ♥ ❞ ♦ ❞ ❡ ❧ ❛ ✐ ♥ ❢ ♦ ♠ ✐ ❝ ❛ ✳ ❆ ♥ ❡ ✐ ♦ ♠ ❡ ♥ ❡ ✱ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❞ ❡ ❝ ♦ ♠ ♣ ✉ ❛ ❞ ♦ ❛ ❡ ❛ ♥ ✐ ♥ ❝ ❧ ✉ ❞ ♦ ❝ ♦ ♠ ♦  

♣ ❛ ❡ ❞ ❡ ❧ ❛ ✈ ❡ ♥ ❛ ❞ ❡ ❧ ❡ ✉ ✐ ♣ ♦ ♦ ❤ ❛ ❞ ✇ ❛ ❡ ② ❡ ❛ ♣ ❡ ♠ ✐ ✐ ❞ ♦ ❡ ❛ ❧ ✐ ③ ❛ ❝ ♦ ♣ ✐ ❛ ❞ ❡ ❧ ♠ ✐ ♠ ♦ ② ❡ ♥ ❡ ❛ ❝ ❝ ❡ ♦  

✐ ♥ ❝ ❧ ✉ ♦ ❛ ❧ ❝ ❞ ✐ ❣ ♦ ❢ ✉ ❡ ♥ ❡ ♣ ❛ ❛ ♣ ♦ ❞ ❡ ❤ ❛ ❝ ❡ ♠ ♦ ❞ ✐ ✜ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ♣ ❛ ❛ ❛ ❞ ❛ ♣ ❛ ❧ ♦ ❛ ❧ ❛ ♥ ❡ ❝ ❡ ✐ ❞ ❛ ❞ ❡ ♣ ❛ ✐ ✲  

❝ ✉ ❧ ❛ ❡ ✳ ❈ ♦ ♥ ❡ ❧ ❛ ❞ ✈ ❡ ♥ ✐ ♠ ✐ ❡ ♥ ♦ ❞ ❡ ❧ ♠ ❡ ❝ ❛ ❞ ♦ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ❛ ✐ ♥ ❢ ♦ ♠ ✐ ❝ ♦ ✱ ♣ ✐ ♥ ❝ ✐ ♣ ❛ ❧ ♠ ❡ ♥ ❡ ✐ ♠ ♣ ✉ ❧ ❛ ❞ ♦  

❛ ✜ ♥ ❛ ❧ ❡ ❞ ❡ ❧ ❛ ❞ ❝ ❛ ❞ ❛ ❞ ❡ ✶ ✾ ✼ ✵ ❝ ♦ ♥ ❡ ❧ ✉ ❣ ✐ ♠ ✐ ❡ ♥ ❞ ♦ ❞ ❡ ▼ ✐ ❝ ♦ ♦ ❢  ➤  ✱ ❡ ❧ ♦ ❢ ✇ ❛ ❡ ❡ ❝ ♦ ♥ ✈ ✐ ✐ ❡ ♥  

✉ ♥ ♣ ♦ ❞ ✉ ❝ ♦ ✉ ❡ ❡ ✈ ❡ ♥ ❞ ❛ ♣ ♦ ❡ ♣ ❛ ❛ ❞ ♦ ✱ ❜ ❛ ❥ ♦ ✉ ♥ ♠ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ♠ ✐ ❡ ♥ ♦ ♣ ❛ ❛ ✉ ✉ ♦ ✱ ❧ ♦ ❝ ✉ ❛ ❧  

✐ ♠ ♣ ❧ ✐ ❝ ❛ ✉ ♥ ❛ ❡ ✐ ❡ ❞ ❡ ❡ ✐ ❝ ❝ ✐ ♦ ♥ ❡ ② ❛ ✉ ❡ ❧ ❛ ❝ ♦ ♠ ♣ ❛ ❛ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❞ ♦ ❛ ♠ ❛ ♥ ✐ ❡ ♥ ❡ ♥ ♦ ❞ ♦ ❧ ♦ ❞ ❡ ✲  

❡ ❝ ❤ ♦ ♦ ❜ ❡ ❡ ❧ ♠ ✐ ♠ ♦ ✱ ❧ ♦ ❝ ✉ ❛ ❧ ✐ ♠ ♣ ✐ ❞ ❡ ❛ ❧ ❝ ♦ ♠ ♣ ❛ ❞ ♦ ❝ ♦ ♣ ✐ ❛ ❧ ♦ ✱ ✐ ♥ ❛ ❧ ❛ ❧ ♦ ❡ ♥ ♦ ❛ ❝ ♦ ♠ ♣ ✉ ❛ ❞ ♦ ❛ ✱  

✈ ❡ ❡ ❧ ❝ ❞ ✐ ❣ ♦ ❢ ✉ ❡ ♥ ❡ ♣ ❛ ❛ ❡ ♥ ❡ ♥ ❞ ❡ ✉ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ♠ ✐ ❡ ♥ ♦ ♦ ❤ ❛ ❝ ❡ ♠ ♦ ❞ ✐ ✜ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ♦ ❛ ❞ ❛ ♣ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡  

❛ ❝ ✉ ❡ ❞ ♦ ❛ ✉ ♣ ♦ ♣ ✐ ❛ ♥ ❡ ❝ ❡ ✐ ❞ ❛ ❞ ❡ ✳

❊ ♥ ✶ ✾ ✽ ✸ ❘ ✐ ❝ ❤ ❛ ❞ ❙ ❛ ❧ ❧ ♠ ❛ ♥  

✷ 

✱ ✉ ♥ ❛ ❝ ❛ ❞ ♠ ✐ ❝ ♦ ❡ ✐ ♥ ❢ ♦ ♠ ✐ ❝ ♦ ❞ ❡ ▼ ■ ❚ ❢ ✉ ♥ ❞ ❛ ❡ ❧ ♠ ♦ ✈ ✐ ♠ ✐ ❡ ♥ ♦ ❞ ❡  

♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ❝ ✉ ② ♦ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ♣ ♦ ❣ ❛ ♠ ❛ ✉ ❡ ♣ ✉ ❡ ❞ ❛ ♥ ❡ ❞ ❡ ❧ ✐ ❜ ❡ ❛ ❝ ❝ ❡ ♦ ♣ ❛ ❛ ✉ ✐ ❡ ♥ ❡  

❞ ❡ ❡ ❡ ♥ ✉ ❛ ❧ ♦ ✳ ▲ ❛ ❝ ❛ ❛ ❝ ❡ ✐ ❝ ❛ ❞ ❡ ✉ ♥ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ♦ ♥ ♣ ✐ ♥ ❝ ✐ ♣ ❛ ❧ ♠ ❡ ♥ ❡ ❧ ❛  

✐ ❣ ✉ ✐ ❡ ♥ ❡ ✿

✉ ❡ ❞ ❡ ❡ ♦ ❜ ❡ ♥ ✐ ❞ ♦ ❡ ♥ ❢ ♦ ♠ ❛ ❣ ❛ ✉ ✐ ❛ ✱ ❛ ✉ ♥ ✉ ❡ ❡ ♦ ♥ ♦ ✐ ❡ ♠ ♣ ❡ ❡ ♦ ❛ ❧ ♠ ❡ ♥ ❡ ❝ ✐ ❡ ♦ ✱ ② ❛ ✉ ❡  

❛ ❧ ❣ ✉ ♥ ❛ ✈ ❡ ❝ ❡ ❡ ♣ ✉ ❡ ❞ ❡ ❝ ♦ ❜ ❛ ❡ ❧ ❝ ♦ ♦ ❞ ❡ ❧ ♠ ❡ ❞ ✐ ♦ ❡ ♥ ❡ ❧ ❝ ✉ ❛ ❧ ❡ ❞ ✐ ✐ ❜ ✉ ② ❛ ✱ ♣ ❡ ♦ ♥ ♦ ❡ ① ✐ ❡ ✉ ♥  

♣ ❛ ❣ ♦ ♣ ♦ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ♣ ❛ ❛ ✉ ✉ ♦ ✳

✉ ❡ ❞ ❡ ❝ ♦ ♣ ✐ ❛ ❡ ❧ ❛ ✈ ❡ ❝ ❡ ✉ ❡ ❡ ❞ ❡ ❡ ❡ ❡ ✐ ♥ ❛ ❧ ❛ ❡ ❡ ♥ ♦ ❞ ❛ ❧ ❛ ❝ ♦ ♠ ♣ ✉ ❛ ❞ ♦ ❛ ✉ ❡ ❡ ❡ ✉ ✐ ❡ ❛ ✱  

✐ ♥ ❧ ✐ ♠ ✐ ❛ ❝ ✐ ♦ ♥ ❡ ♥ ✐ ❡ ✐ ❝ ❝ ✐ ♥ ❛ ❧ ❣ ✉ ♥ ❛ ✳

❙ ❡ ♣ ✉ ❡ ❞ ❡ ❡ ♥ ❡ ❛ ❝ ❝ ❡ ♦ ❛ ❧ ❝ ❞ ✐ ❣ ♦ ❢ ✉ ❡ ♥ ❡ ♣ ❛ ❛ ♣ ♦ ❞ ❡ ❡ ✉ ❞ ✐ ❛ ② ❝ ♦ ♠ ♣ ❡ ♥ ❞ ❡ ✉ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ♠ ✐ ❡ ♥ ♦ ✳

❚ ❛ ♠ ❜ ✐ ♥ ♣ ✉ ❡ ❞ ❡ ♠ ♦ ❞ ✐ ✜ ❝ ❛ ❡ ✐ ❡ ♥ ❡ ❝ ❡ ❛ ✐ ♦ ♣ ❛ ❛ ❛ ❞ ❛ ♣ ❛ ❧ ♦ ❛ ❞ ✐ ✐ ♥ ❛ ♥ ❡ ❝ ❡ ✐ ❞ ❛ ❞ ❡ ✳

❊ ❡ ❧ ✐ ♠ ♦ ❛ ♣ ❡ ❝ ♦ ❡ ❡ ♣ ❡ ❝ ✐ ❛ ❧ ♠ ❡ ♥ ❡ ✐ ♠ ♣ ♦ ❛ ♥ ❡ ❡ ♥ ❡ ❧ ❝ ❛ ♦ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ❞ ❡ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ❡ ♥ ✜ ❝ ♦  

❞ ❡ ❜ ✐ ❞ ♦ ❛ ✉ ❡ ❡ ♥ ❡ ❧ ♣ ♦ ❝ ❡ ♦ ❞ ❡ ✐ ♥ ✈ ❡ ✐ ❣ ❛ ❝ ✐ ♥ ② ❞ ❡ ❛ ♦ ❧ ❧ ♦ ✱ ❛ ❝ ♦ ♠ ♦ ❡ ♥ ❧ ❛ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ✱  

♥ ♦ ❡ ✈ ❧ ✐ ❞ ♦ ❡ ❧ ♠ ♦ ❞ ❡ ❧ ♦ ❞ ❡ ✏ ❝ ❛ ❥ ❛ ♥ ❡ ❣ ❛ ✑ ✉ ✐ ❧ ✐ ③ ❛ ❞ ♦ ♣ ♦ ❡ ❧ ♦ ❢ ✇ ❛ ❡ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ✱ ❞ ♦ ♥ ❞ ❡ ❡ ❛ ❝ ❡ ♣ ❛ ♥ ❧ ♦  

❡ ✉ ❧ ❛ ❞ ♦ ♦ ❜ ❡ ♥ ✐ ❞ ♦ ✐ ♥ ✈ ❛ ❧ ✐ ❞ ❛ ❧ ❛ ❢ ♦ ♠ ❛ ❞ ❡ ♦ ❜ ❡ ♥ ❡ ❧ ♦ ✱ ♣ ♦ ❧ ♦ ✉ ❡ ❡ ✉ ♠ ❛ ♠ ❡ ♥ ❡ ✐ ♠ ♣ ♦ ❛ ♥ ❡  

❡ ♥ ❡ ♥ ❞ ❡ ❧ ♦ ❛ ❧ ❣ ♦ ✐ ♠ ♦ ✉ ❡ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ✉ ✐ ❧ ✐ ③ ❛ ♥ ✳

❈ ♦ ♥ ❧ ❛ ❧ ❧ ❡ ❣ ❛ ❞ ❛ ② ❡ ① ♣ ❛ ♥ ✐ ♥ ❞ ❡ ✐ ♥ ❡ ♥ ❡ ❛ ♥ ✐ ✈ ❡ ❧ ❣ ❧ ♦ ❜ ❛ ❧ ✱ ❡ ❧ ♠ ♦ ✈ ✐ ♠ ✐ ❡ ♥ ♦ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ❡ ❤ ❛  

❢ ♦ ❛ ❧ ❡ ❝ ✐ ❞ ♦ ② ❡ ① ❡ ♥ ❞ ✐ ❞ ♦ ❞ ❡ ❛ ❧ ❢ ♦ ♠ ❛ ✉ ❡ ❡ ♥ ❧ ❛ ❛ ❝ ✉ ❛ ❧ ✐ ❞ ❛ ❞ ✱ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❞ ♦ ❜ ❛ ❥ ♦ ❡ ❡  

♠ ♦ ❞ ❡ ❧ ♦ ✱ ❝ ♦ ♠ ♣ ✐ ❡ ♥ ❝ ♦ ♥ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ❡ ❡ ♥ ❝ ❛ ❧ ✐ ❞ ❛ ❞ ✱ ✐ ♥ ❝ ❧ ✉ ♦ ❡ ♥ ❛ ❧ ❣ ✉ ♥ ♦ ❝ ❛ ♦ ✉ ♣ ❡ ❛ ♥ ❞ ♦  

✶ 

❤ ♣ ✿ ✴ ✴ ❡ ♥ ✳ ✇ ✐ ❦ ✐ ♣ ❡ ❞ ✐ ❛ ✳ ♦ ❣ ✴ ✇ ✐ ❦ ✐ ✴ ❋ ❡ ❡ ❴ ❛ ♥ ❞ ❴ ♦ ♣ ❡ ♥ ✲ ♦ ✉ ❝ ❡ ❴ ♦ ❢ ✇ ❛ ❡  

✷ 

❤ ♣ ✿ ✴ ✴ ❡ ✳ ✇ ✐ ❦ ✐ ♣ ❡ ❞ ✐ ❛ ✳ ♦ ❣ ✴ ✇ ✐ ❦ ✐ ✴ ❘ ✐ ❝ ❤ ❛ ❞ ❴ ❙ ❛ ❧ ❧ ♠ ❛ ♥  

✶ ✷ 

8/19/2019 Notas Optimizacion Ingenieros-libre

13/86

✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙  

❛ ✉ ♣ ❛ ❡ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ❡ ✳ ▼ ✉ ❝ ❤ ❛ ♦ ❣ ❛ ♥ ✐ ③ ❛ ❝ ✐ ♦ ♥ ❡ ❡ ♥ ♦ ❞ ♦ ❡ ❧ ♠ ✉ ♥ ❞ ♦ ❤ ❛ ♥ ❛ ❞ ♦ ♣ ❛ ❞ ♦ ❡ ❧ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡  

♣ ♦ ❞ ✐ ✈ ❡ ❛ ❛ ③ ♦ ♥ ❡ ✱ ✐ ❡ ♥ ❞ ♦ ✉ ♥ ❛ ❞ ❡ ❡ ❧ ❧ ❛ ❡ ❧ ❛ ❤ ♦ ♦ ❡ ♥ ❧ ♦ ❝ ♦ ♦ ❞ ❡ ❧ ✐ ❝ ❡ ♥ ❝ ✐ ❛ ✱ ♣ ❡ ♦ ❡ ♥ ♠ ✉ ❝ ❤ ♦ ❝ ❛ ♦  

♥ ♦ ❡ ❧ ❛ ♣ ✐ ♥ ❝ ✐ ♣ ❛ ❧ ✱ ✐ ♥ ♦ ✉ ❡ ❡ ❤ ❛ ❝ ♦ ♠ ♣ ♦ ❜ ❛ ❞ ♦ ❧ ❛ ❝ ❛ ❧ ✐ ❞ ❛ ❞ ② ✈ ❡ ❛ ✐ ❧ ✐ ❞ ❛ ❞ ❞ ❡ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❧ ✐ ❜ ❡  

♣ ♦ ❡ ♥ ❝ ✐ ♠ ❛ ❞ ❡ ❧ ♦ ❢ ✇ ❛ ❡ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ✳

❊ ♥ ❡ ❡ ❝ ✉ ♦ ❡ ❛ ❡ ♠ ♦ ✉ ✐ ❧ ✐ ③ ❛ ♥ ❞ ♦ ✈ ❛ ✐ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ✉ ❡ ♣ ❡ ❡ ♥ ❡ ❝ ❡ ♥ ❛ ❧ ❛ ❝ ❛ ❡ ❣ ♦ ❛ ❞ ❡ ♦ ❢ ✇ ❛ ❡  

❧ ✐ ❜ ❡ ✱ ❧ ♦ ❝ ✉ ❛ ❧ ♣ ❡ ❡ ♥ ❛ ✉ ♥ ❛ ❡ ✐ ❡ ❞ ❡ ✈ ❡ ♥ ❛ ❥ ❛ ♣ ❛ ❛ ❡ ❧ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ❧ ♣ ♦ ❣ ❛ ♠ ❛ ❞ ❡ ❧ ❝ ✉ ♦ ✳ ❊ ♥ ❡ ❧ ❛  

✈ ❡ ♥ ❛ ❥ ❛ ♣ ♦ ❞ ❡ ♠ ♦ ♠ ❡ ♥ ❝ ✐ ♦ ♥ ❛ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✿

❚ ♦ ❞ ♦ ❧ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❛ ✉ ✐ ❧ ✐ ③ ❛ ♣ ✉ ❡ ❞ ❡ ♥ ❡ ❞ ❡ ❝ ❛ ❣ ❛ ❞ ♦ ❞ ❡ ✐ ♥ ❡ ♥ ❡ ✐ ♥ ♥ ✐ ♥ ❣ ♥ ❝ ♦ ♦ ✳

✉ ❡ ❞ ❡ ♥ ✐ ♥ ❛ ❧ ❛ ❡ ❡ ♥ ❧ ❛ ❝ ♦ ♠ ♣ ✉ ❛ ❞ ♦ ❛ ♣ ❡ ♦ ♥ ❛ ❧ ❡ ❞ ❡ ❧ ♦ ❡ ✉ ❞ ✐ ❛ ♥ ❡ ✱ ❧ ♦ ✉ ❡ ❧ ❡ ♣ ❡ ♠ ✐ ❡  

❡ ♥ ❡ ❧ ♦ ✐ ❡ ♠ ♣ ❡ ❞ ✐ ♣ ♦ ♥ ✐ ❜ ❧ ❡ ♣ ❛ ❛ ❡ ❛ ❧ ✐ ③ ❛ ♣ ❝ ✐ ❝ ❛ ② ❛ ❡ ❛ ✳ ▲ ❛ ♠ ❛ ② ♦ ❛ ❞ ❡ ❡ ❧ ❧ ♦ ♣ ✉ ❡ ❞ ❡ ❡  

✐ ♥ ❛ ❧ ❛ ❞ ♦ ❡ ♥ ❞ ✐ ✈ ❡ ♦ ✐ ❡ ♠ ❛ ♦ ♣ ❡ ❛ ✐ ✈ ♦ ❝ ♦ ♠ ♦ ▼ ❙ ❲ ✐ ♥ ❞ ♦ ✇  ➤  ✱ ▲ ✐ ♥ ✉ ① ♦ ▼ ❛ ❝ ❖ ❙  ➤ 

✉ ❡ ❞ ❡ ♥ ✉ ✐ ❧ ✐ ③ ❛ ❧ ♦ ✐ ♥ ❡ ✐ ❝ ❝ ✐ ♥ ✱ ✐ ♥ ❝ ❧ ✉ ♦ ♣ ❛ ❛ ✜ ♥ ❡ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ❡ ♣ ♦ ❧ ♦ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ♥ ❛ ♣ ❧ ✐ ❝ ❛ ❧ ♦  

❡ ♥ ✉ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ♣ ♦ ❢ ❡ ✐ ♦ ♥ ❛ ❧ ✳

❊ ① ✐ ❡ ✉ ♥ ❛ ❣ ❛ ♥ ❝ ❛ ♥ ✐ ❞ ❛ ❞ ❞ ❡ ♠ ❛ ❡ ✐ ❛ ❧ ❞ ❡ ❡ ❢ ❡ ❡ ♥ ❝ ✐ ❛ ❞ ✐ ♣ ♦ ♥ ✐ ❜ ❧ ❡ ❡ ♥ ✐ ♥ ❡ ♥ ❡ ♣ ❛ ❛ ❛ ♣ ❡ ♥ ❞ ❡ ❛  

✉ ✐ ❧ ✐ ③ ❛ ❧ ♦ ✳

▲ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ♥ ❡ ♥ ❝ ♦ ♥ ❛ ♥ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ♣ ♦ ♣ ❛ ❡ ❞ ❡ ✉ ♥ ❛ ❝ ♦ ♠ ✉ ♥ ✐ ❞ ❛ ❞ ♠ ✉ ②  

❛ ❝ ✐ ✈ ❛ ❛ ❧ ❡ ❞ ❡ ❞ ♦ ❞ ❡ ❧ ♠ ✉ ♥ ❞ ♦ ✱ ♣ ♦ ❧ ♦ ✉ ❡ ❢ ❡ ❝ ✉ ❡ ♥ ❡ ♠ ❡ ♥ ❡ ❡ ❧ ✐ ❜ ❡ ❛ ♥ ✈ ❡ ✐ ♦ ♥ ❡ ❛ ❝ ✉ ❛ ❧ ✐ ③ ❛ ❞ ❛ ②  

♠ ❡ ❥ ♦ ❛ ❞ ❛ ✳

✸ ✳ ✷ ✳ ❙ ♦ ❢ ✇ ❛ ❡ ❛ ✉ ✐ ❧ ✐ ③ ❛  

❊ ♥ ❡ ❛ ❡ ❝ ❝ ✐ ♥ ❡ ❡ ♥ ✉ ♠ ❡ ❛ ❝ ❛ ❞ ❛ ✉ ♥ ❛ ❞ ❡ ❧ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ♥ ✉ ✐ ❧ ✐ ③ ❛ ❡  

♣ ❛ ❛ ❡ ♦ ❧ ✈ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ② ❡ ❤ ❛ ❝ ❡ ✉ ♥ ❛ ❜ ❡ ✈ ❡ ❡ ❡ ❛ ❞ ❡ ❡ ❧ ❧ ❛ ✱ ✐ ♥ ❞ ✐ ❝ ❛ ♥ ❞ ♦ ✉ ♣ ✐ ♥ ❝ ✐ ♣ ❛ ❧  

❡ ❛ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ❛ ❝ ♦ ♠ ♦ ❛ ❧ ❣ ✉ ♥ ❛ ❞ ❡ ✉ ❝ ❛ ❛ ❝ ❡ ✐ ❝ ❛ ♠ ❛ ✐ ♠ ♣ ♦ ❛ ♥ ❡ ✳ ❈ ✉ ❛ ♥ ❞ ♦ ❡ ♣ ❡ ✐ ♥ ❡ ♥ ❡  

❡ ✐ ♥ ❞ ✐ ❝ ❛ ❡ ❧ ♥ ♦ ♠ ❜ ❡ ❞ ❡ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ✐ ♠ ✐ ❧ ❛ ♣ ❛ ❛ ✉ ❛ ❧ ♦ ❝ ♦ ♠ ♦ ❡ ❢ ❡ ❡ ♥ ❝ ✐ ❛ ② ♣ ✉ ♥ ♦ ❞ ❡  

❝ ♦ ♠ ♣ ❛ ❛ ❝ ✐ ♥ ❡ ♥ ❝ ✉ ❛ ♥ ♦ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❧ ✐ ❞ ❛ ❞ ✳ ❊ ♣ ♦ ✐ ❜ ❧ ❡ ✉ ❡ ♥ ♦ ♦ ❞ ❛ ❡ ❛ ♥ ✉ ✐ ❧ ✐ ③ ❛ ❞ ❛ ❡ ♥ ❡ ❧ ❝ ✉ ♦ ✱ ♣ ❡ ♦  

❡ ✐ ♠ ♣ ♦ ❛ ♥ ❡ ❝ ♦ ♥ ♦ ❝ ❡ ✉ ❡ ① ✐ ❡ ♥ ❝ ✐ ❛ ② ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ♣ ❛ ❛ ✉ ✐ ❡ ♥ ❡ ❞ ❡ ❡ ❡ ♥ ❡ ♥ ✉ ♥ ❢ ✉ ✉ ♦ ♣ ♦ ❢ ✉ ♥ ❞ ✐ ③ ❛  

♦ ❜ ❡ ❛ ❧ ❣ ✉ ♥ ❛ ❞ ❡ ❡ ❧ ❧ ❛ ✳

✸ ✳ ✷ ✳ ✶ ✳ ▼ ❛ ① ✐ ♠ ❛  

▼ ❛ ① ✐ ♠ ❛ ♣ ✉ ❡ ❞ ❡ ❡ ❝ ❛ ❛ ❧ ♦ ❣ ❛ ❞ ♦ ❞ ❡ ♥ ♦ ❞ ❡ ❧ ❛ ❝ ❛ ❡ ❣ ♦ ❛ ❞ ❡ ❧ ♦ ❈ ❆ ❙ ✭ ❈ ♦ ♠ ♣ ✉ ❡ ❆ ❧ ❣ ❡ ❜ ❛ ❙ ② ❡ ♠ ✮ ✳

❯ ♥ ❈ ❆ ❙ ❡ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ❛ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ❝ ✉ ② ❛ ❢ ♦ ❛ ❧ ❡ ③ ❛ ❛ ❞ ✐ ❝ ❛ ❡ ♥ ❧ ❛ ❝ ❛ ♣ ❛ ❝ ✐ ❞ ❛ ❞ ❞ ❡ ❡ ❛ ❧ ✐ ③ ❛ ♦ ♣ ❡ ❛ ❝ ✐ ♦ ✲  

♥ ❡ ② ♠ ❛ ♥ ✐ ♣ ✉ ❧ ❛ ❝ ✐ ♥ ♦ ❜ ❡ ♦ ❜ ❥ ❡ ♦ ✐ ♠ ❜ ❧ ✐ ❝ ♦ ✳ ❊ ❞ ❡ ❝ ✐ ✱ ✉ ♥ ❈ ❆ ❙ ❡ ❝ ❛ ♣ ❛ ③ ❞ ❡ ♦ ♠ ❛ ✉ ♥ ❛ ❡ ① ♣ ❡ ✐ ♥  

❡ ♣ ❡ ❡ ♥ ❛ ❞ ❛ ❡ ♥ ❢ ♦ ♠ ❛ ✐ ♠ ❜ ❧ ✐ ❝ ❛ ② ❡ ❛ ❧ ✐ ③ ❛ ♦ ♣ ❡ ❛ ❝ ✐ ♦ ♥ ❡ ❜ ✐ ❝ ❛ ✭ ✉ ♠ ❛ ✱ ♠ ✉ ❧ ✐ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ✱ ❡ ① ♣ ♦ ♥ ❡ ♥ ✲  

❝ ✐ ❛ ❝ ✐ ♥ ✱ ❡ ❝ ✳ ✮ ② ❛ ♠ ❜ ✐ ♥ ♦ ♣ ❡ ❛ ❝ ✐ ♦ ♥ ❡ ♠ ❛ ✈ ❛ ♥ ③ ❛ ❞ ❛ ❝ ♦ ♠ ♦ ❞ ❡ ✐ ✈ ❛ ❝ ✐ ♥ ❡ ✐ ♥ ❡ ❣ ❛ ❝ ✐ ♥ ✐ ♠ ❜ ❧ ✐ ❝ ❛ ✱  

❡ ♣ ❡ ❡ ♥ ❛ ❝ ✐ ♥ ❣ ✜ ❝ ❛ ✱ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ❡ ❝ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ✱ ❡ ❝ ✳

▼ ❛ ① ✐ ♠ ❛ ❞ ❡ ❝ ✐ ❡ ♥ ❞ ❡ ❞ ❡ ❧ ♣ ♦ ❣ ❛ ♠ ❛ ♠ ❛ ♥ ❣ ✉ ♦ ❞ ❡ ♥ ♦ ❞ ❡ ❧ ♦ ❈ ❆ ❙ ✱ ❧ ❧ ❛ ♠ ❛ ❞ ♦ ▼ ❛ ❝ ② ♠ ❛ ✱ ✉ ❡ ❢ ✉  

❞ ❡ ❛ ♦ ❧ ❧ ❛ ❞ ♦ ❡ ♥ ❧ ❛ ❞ ❝ ❛ ❞ ❛ ❞ ❡ ✶ ✾ ✻ ✵ ✳ ❊ ✉ ♥ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ✉ ♠ ❛ ♠ ❡ ♥ ❡ ❝ ♦ ♠ ♣ ❧ ❡ ❛ ✱ ♣ ♦ ❞ ❡ ♦ ❛ ✱ ✈ ❡ ✐ ❧  

② ❡ ❛ ❜ ❧ ❡ ✱ ❛ ♣ ❛ ❡ ❞ ❡ ❡ ✉ ♥ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ✶ ✵ ✵ ✪ ❧ ✐ ❜ ❡ ✳ ❉ ❡ ♥ ♦ ❞ ❡ ❧ ❡ ♠ ❛ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✱ ♣ ✉ ❡ ❞ ❡ ❡  

❛ ♣ ❧ ✐ ❝ ❛ ❞ ♦ ❡ ♥ ❧ ♦ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❝ ❛ ♦ ✿

❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ◆ ♦ ▲ ✐ ♥ ❡ ❛ ❧ ② ◆ ♦ ❘ ❡ ✐ ♥ ❣ ✐ ❞ ❛ ♣ ❛ ❛ ✉ ♥ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡  

❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ▲ ✐ ♥ ❡ ❛ ❧ ❘ ❡ ✐ ♥ ❣ ✐ ❞ ❛ ▼ ✉ ❧ ✐ ✈ ❛ ✐ ❛ ❞ ❛  

❖ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♦ ♥ ◆ ♦ ▲ ✐ ♥ ❡ ❛ ❧ ❘ ❡ ✐ ♥ ❣ ✐ ❞ ❛ ▼ ✉ ❧ ✐ ✈ ❛ ✐ ❛ ❞ ❛  

✶ ✸ 

8/19/2019 Notas Optimizacion Ingenieros-libre

14/86

✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙  

❋ ✐ ❣ ✉ ❛ ✸ ✳ ✶ ✳ ✿ ■ ♠ ❛ ❣ ❡ ♥ ❡ ❞ ❡ ▼ ❛ ① ✐ ♠ ❛  

▼ ❛ ① ✐ ♠ ❛ ❡ ✐ ♠ ✐ ❧ ❛ ❡ ♥ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❧ ✐ ❞ ❛ ❞ ❡ ❛ ❛ ❧ ❣ ✉ ♥ ♦ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ❡ ❛ ❧ ❡ ❝ ♦ ♠ ♦ ▼ ❛ ❤ ❡ ♠ ❛ ✐ ✲  

❝ ❛  ➤ 

✸ 

✱ ▼ ❛ ♣ ❧ ❡  ➤ 

✹ 

♦ ▼ ❛ ❤ ❈ ❛ ❞  ➤ 

✺ 

✳

▼ ❛ ① ✐ ♠ ❛ ♣ ✉ ❡ ❞ ❡ ❡ ❞ ❡ ❝ ❛ ❣ ❛ ❞ ♦ ❡ ♥ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❞ ✐ ❡ ❝ ❝ ✐ ♥ ✿

❤ ♣ ✿ ✴ ✴ ♠ ❛ ① ✐ ♠ ❛ ✳ ♦ ✉ ❝ ❡ ❢ ♦ ❣ ❡ ✳ ♥ ❡ ✴  

✸ ✳ ✷ ✳ ✷ ✳ ❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ❈ ❛ ❧ ❝ ✴ ▲ ✐ ❜ ❡ ❖ ✣ ❝ ❡ ❈ ❛ ❧ ❝  

❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ❡ ✉ ♥ ❛ ✉ ✐ ❡ ❞ ❡ ♦ ✜ ❝ ✐ ♥ ❛ ✉ ❡ ❝ ♦ ♠ ♣ ✐ ❡ ❝ ♦ ♥ ❧ ❛ ♦ ♠ ♥ ✐ ♣ ❡ ❡ ♥ ❡ ▼ ✐ ❝ ♦ ♦ ❢ ❖ ✣ ❝ ❡  ➤  ✳ ❉ ❡ 

❤ ❡ ❝ ❤ ♦ ✱ ❡ ① ✐ ❡ ✉ ♥ ❛ ❧ ♦ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ❝ ♦ ♠ ♣ ❛ ✐ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ❡ ♥ ❡ ❡ ❧ ❧ ❛ ❞ ❡ ♠ ❛ ♥ ❡ ❛ ✉ ❡ ❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ♣ ✉ ❡ ❞ ❡ ❛ ❜ ✐  

❛ ❝ ❤ ✐ ✈ ♦ ❞ ❡ ❖ ✣ ❝ ❡ ✳ ❆ ✉ ♥ ✉ ❡ ❧ ❛ ✉ ✐ ❡ ❡ ❝ ♦ ♠ ♣ ♦ ♥ ❡ ❞ ❡ ✉ ♥ ❛ ❡ ✐ ❡ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ✭ ♣ ♦ ❝ ❡ ❛ ❞ ♦ ❞ ❡ ❡ ① ♦ ✱  

❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❞ ❡ ♣ ❡ ❡ ♥ ❛ ❝ ✐ ♦ ♥ ❡ ✱ ❜ ❛ ❡ ❞ ❡ ❞ ❛ ♦ ✱ ❡ ❝ ✳ ✮ ♥ ♦ ✐ ♥ ❡ ❡ ❛ ♣ ✐ ♥ ❝ ✐ ♣ ❛ ❧ ♠ ❡ ♥ ❡ ✉ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ❞ ❡  

❍ ♦ ❥ ❛ ❊ ❧ ❡ ❝ ♥ ✐ ❝ ❛ ✱ ❧ ❧ ❛ ♠ ❛ ❞ ❛ ❈ ❛ ❧ ❝ ✱ ♠ ✉ ② ✐ ♠ ✐ ❧ ❛ ② ❝ ♦ ♠ ♣ ❛ ✐ ❜ ❧ ❡ ❝ ♦ ♥ ❊ ① ❝ ❡ ❧ ❞ ❡ ▼ ❙ ❖ ✣ ❝ ❡ ✱ ♣ ♦ ❧ ♦ ✉ ❡  

❧ ♦ ♣ ♦ ❝ ❡ ❞ ✐ ♠ ✐ ❡ ♥ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ♣ ✉ ❡ ❞ ❡ ♥ ❢ ❝ ✐ ❧ ♠ ❡ ♥ ❡ ② ❝ ♦ ♥ ♣ ♦ ❝ ♦ ❝ ❛ ♠ ❜ ✐ ♦ ✱ ❡ ♣ ❧ ✐ ❝ ❛ ❡ ❞ ❡ ✉ ♥ ❛  

❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❛ ♦ ❛ ✳

❊ ① ✐ ❡ ♥ ✷ ♦ ♣ ❝ ✐ ♦ ♥ ❡ ♣ ❛ ❛ ❡ ❝ ♦ ❣ ❡ ❡ ♥ ❡ ❧ ❝ ❛ ♦ ❞ ❡ ❈ ❛ ❧ ❝ ✳ ▲ ❛ ♣ ✐ ♠ ❡ ❛ ❡ ❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ✭ ❛ ♥ ❡ ✐ ♦ ♠ ❡ ♥ ❡  

❧ ❧ ❛ ♠ ❛ ❞ ❛ ❙ ❛ ❖ ✣ ❝ ❡ ✮ ✱ ✉ ❡ ❡ ❧ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ♠ ❛ ♥ ✐ ❣ ✉ ❛ ✱ ❝ ♦ ♥ ❝ ❛ ✐ ✷ ✵ ❛ ♦ ❞ ❡ ❡ ① ✐ ❡ ♥ ❝ ✐ ❛ ② ✉ ❡ ❤ ❛  

❛ ❧ ❝ ❛ ♥ ③ ❛ ❞ ♦ ✉ ♥ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ♠ ❛ ❞ ✉ ❡ ③ ② ❡ ❛ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ✉ ❡ ♣ ❡ ♠ ✐ ❡ ❡ ❛ ✉ ✐ ❧ ✐ ③ ❛ ❞ ♦ ❡ ♥ ♠ ✉ ❝ ❤ ❛ ♦ ❣ ❛ ♥ ✐ ③ ❛ ❝ ♦ ♥ ❡  

❡ ♥ ❡ ❧ ♠ ✉ ♥ ❞ ♦ ❝ ♦ ♠ ♦ ✉ ♥ ✉ ✐ ✉ ♦ ❞ ❡ ❊ ① ❝ ❡ ❧ ✳ ▲ ❛ ❡ ❣ ✉ ♥ ❞ ❛ ♦ ♣ ❝ ✐ ♥ ❡ ▲ ✐ ❜ ❡ ❖ ✣ ❝ ❡ ✱ ✉ ❡ ❡ ✉ ♥ ❛ ❞ ❡ ✐ ✈ ❛ ❝ ✐ ♥  

✉ ❡ ❤ ✐ ③ ♦ ✉ ♥ ❣ ✉ ♣ ♦ ❞ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❞ ♦ ❡ ❡ ♥ ❡ ❧ ✷ ✵ ✶ ✵ ❛ ♣ ❛ ✐ ❞ ❡ ❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ❝ ✉ ❛ ♥ ❞ ♦ ❡ ❢ ✉ ❛ ❞ ✉ ✐ ✐ ❞ ♦  

♣ ♦ ❖ ❛ ❝ ❧ ❡ ✳ ♦ ❛ ❛ ❡ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ✱ ❡ ❧ ❝ ❞ ✐ ❣ ♦ ❢ ✉ ❡ ♥ ❡ ♦ ✐ ❣ ✐ ♥ ❛ ❧ ❞ ❡ ▲ ✐ ❜ ❡ ❖ ✣ ❝ ❡  

❡ ❛ ❡ ❧ ♠ ✐ ♠ ♦ ✉ ❡ ❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ♣ ♦ ❧ ♦ ✉ ❡ ❛ ♠ ❜ ♦ ♦ ♥ ❝ ❛ ✐ ✐ ❞ ♥ ✐ ❝ ♦ ✱ ❧ ♦ ✉ ❡ ✐ ♠ ♣ ❧ ✐ ❝ ❛ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ✉ ❛ ❡  

✉ ♥ ♦ ✉ ♦ ♦ ✐ ♥ ❞ ✐ ✐ ♥ ❛ ♠ ❡ ♥ ❡ ✳

✸ 

❤ ♣ ✿ ✴ ✴ ✇ ✇ ✇ ✳ ✇ ♦ ❧ ❢ ❛ ♠ ✳ ❝ ♦ ♠ ✴  

✹ 

❤ ♣ ✿ ✴ ✴ ✇ ✇ ✇ ✳ ♠ ❛ ♣ ❧ ❡ ♦ ❢ ✳ ❝ ♦ ♠ ✴ ♣ ♦ ❞ ✉ ❝ ✴ ♠ ❛ ♣ ❧ ❡ ✴  

✺ 

❤ ♣ ✿ ✴ ✴ ✇ ✇ ✇ ✳ ♣ ❝ ✳ ❝ ♦ ♠ ✴ ♣ ♦ ❞ ✉ ❝ ✴ ♠ ❛ ❤ ❝ ❛ ❞ ✴  

✶ ✹ 

8/19/2019 Notas Optimizacion Ingenieros-libre

15/86

✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙  

❋ ✐ ❣ ✉ ❛ ✸ ✳ ✷ ✳ ✿ ❙ ♦ ❧ ✈ ❡ ❞ ❡ ❈ ❛ ❧ ❝  

♦ ❛ ❛ ❡ ❞ ❡ ✉ ♥ ❛ ❤ ♦ ❥ ❛ ❡ ❧ ❡ ❝ ♥ ✐ ❝ ❛ ✱ ❈ ❛ ❧ ❝ ♣ ✉ ❡ ❞ ❡ ✉ ✐ ❧ ✐ ③ ❛ ❡ ♣ ❛ ❛ ✉ ♥ ✐ ♥ ❢ ♥ ❞ ❡ ♣ ♦ ♣ ✐ ♦ ❝ ♦ ♠ ♦  

❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❞ ❡ ❝ ❧ ❝ ✉ ❧ ♦ ✳ ❊ ♥ ❡ ❧ ❝ ❛ ♦ ❞ ❡ ❧ ❛ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✱ ❧ ❛ ✸ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❛ ✉ ♠ ❡ ♥ ❝ ✐ ♦ ♥ ❛ ❞ ❛ ❝ ✉ ❡ ♥ ❛ ♥  

❝ ♦ ♥ ✉ ♥ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❞ ❡ ♥ ♦ ♠ ✐ ♥ ❛ ❞ ❛  ✏ ♦ ❧ ✈ ❡ ✑  

✉ ❡ ♣ ❡ ♠ ✐ ❡ ❡ ♦ ❧ ✈ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❞ ❡ ❞ ✐ ✐ ♥ ❛ ♥ ❞ ♦ ❧ ❡ ✳

❉ ❛ ❞ ♦ ✉ ❡ ♠ ✉ ❝ ❤ ❛ ♣ ❡ ♦ ♥ ❛ ❡ ♥ ❢ ❛ ♠ ✐ ❧ ✐ ❛ ✐ ③ ❛ ❞ ❛ ❝ ♦ ♥ ❡ ❧ ✉ ♦ ❞ ❡ ❤ ♦ ❥ ❛ ❡ ❧ ❡ ❝ ♥ ✐ ❝ ❛ ✱ ❛ ❝ ♦ ♠ ♦ ✉  

❡ ✉ ❝ ✉ ❛ ♠ ❛ ✐ ❝ ✐ ❛ ❧ ❞ ❡ ❝ ❡ ❧ ❞ ❛ ✱ ❡ ❤ ❛ ❝ ❡ ♠ ✉ ② ❡ ♥ ❝ ✐ ❧ ❧ ♦ ❡ ❧ ♣ ❧ ❛ ♥ ❡ ❛ ♠ ✐ ❡ ♥ ♦ ❞ ❡ ♠ ♦ ❞ ❡ ❧ ♦ ✱ ❡ ♣ ❡ ❝ ✐ ❛ ❧ ♠ ❡ ♥ ❡  

❡ ♥ ❡ ❧ ❡ ♠ ❛ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ❞ ❡ ❡ ❞ ❡ ✱ ❡ ♠ ❛ ❞ ❡ ♠ ✉ ❝ ❤ ❛ ✐ ♠ ♣ ♦ ❛ ♥ ❝ ✐ ❛ ❡ ♥ ❧ ❛ ✐ ♥ ❣ ❡ ♥ ✐ ❡ ❛ ✈ ✐ ❛ ❧ ✳

❖ ♣ ❡ ♥ ❖ ✣ ❝ ❡ ♣ ✉ ❡ ❞ ❡ ❡ ❞ ❡ ❝ ❛ ❣ ❛ ❞ ♦ ❡ ♥ ✿

❤ ♣ ✿ ✴ ✴ ✇ ✇ ✇ ✳ ♦ ♣ ❡ ♥ ♦ ❢ ❢ ✐ ❝ ❡ ✳ ♦ ❣ ✴  

▲ ✐ ❜ ❡ ❖ ✣ ❝ ❡ ♣ ✉ ❡ ❞ ❡ ❡ ❞ ❡ ❝ ❛ ❣ ❛ ❞ ♦ ❡ ♥ ✿

❤ ♣ ✿ ✴ ✴ ✇ ✇ ✇ ✳ ❧ ✐ ❜ ❡ ♦ ❢ ❢ ✐ ❝ ❡ ✳ ♦ ❣ ✴  

✸ ✳ ✷ ✳ ✸ ✳ ❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦  

❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦ ❡ ✉ ♥ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❝ ♦ ♥ ❝ ✐ ❡ ♦ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ✐ ♠ ✐ ❧ ✐ ✉ ❞ ❝ ♦ ♥ ❧ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧  

▼ ❛ ❤ ❈ ❛ ❞  ➤  ♠ ❡ ♥ ❝ ✐ ♦ ♥ ❛ ❞ ❛ ❛ ♥ ❡ ✐ ♦ ♠ ❡ ♥ ❡ ✳ ❙ ❡ ❛ ❛ ❞ ❡ ✉ ♥ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ❞ ❡ ❝ ❧ ❝ ✉ ❧ ♦ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ✐ ♥ ✲  

❡ ❛ ❝ ✐ ✈ ❛ ✉ ❡ ♣ ❡ ♠ ✐ ❡ ♣ ❧ ❛ ♠ ❛ ❡ ① ♣ ❡ ✐ ♦ ♥ ❡ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ② ♦ ❜ ❡ ♥ ❡ ❧ ♦ ❡ ✉ ❧ ❛ ❞ ♦ ✉ ✐ ❧ ✐ ③ ❛ ♥ ❞ ♦ ❧ ❛  

✐ ♠ ❜ ♦ ❧ ♦ ❣ ❛ ❝ ♥ ✐ ❝ ❛ ❛ ❧ ② ❝ ♦ ♠ ♦ ❡ ♣ ❡ ❡ ♥ ❛ ❡ ♥ ❧ ❛ ♠ ❛ ② ♦ ❛ ❞ ❡ ❧ ✐ ❜ ♦ ❞ ❡ ❡ ① ♦ ② ❝ ♦ ♠ ♦ ✐ ❡ ❡ ✉ ✈ ✐ ❡ ✲  

❡ ❛ ❜ ❛ ❥ ❛ ♥ ❞ ♦ ♦ ❜ ❡ ✉ ♥ ❛ ❤ ♦ ❥ ❛ ❞ ❡ ♣ ❛ ♣ ❡ ❧ ✳ ❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❜ ❛ ❥ ♦ ❡ ❧ ❝ ♦ ♥ ❝ ❡ ♣ ♦ ❞ ❡   ❡ ❣ ✐ ♦ ♥ ❞ ❡  

❝ ❧ ❝ ✉ ❧ ♦  

✱ ❡ ❞ ❡ ❝ ✐ ✉ ❡ ❡ ❧ ❡ ✉ ❧ ❛ ❞ ♦ ❞ ❡ ✉ ♥ ❛ ❡ ① ♣ ❡ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ❞ ❡ ❧ ❡ ❛ ❞ ♦ ♥ ❞ ❡ ❞ ✐ ❝ ❤ ❛ ❡ ① ♣ ❡ ✐ ♥ ❡  

❡ ♥ ❝ ✉ ❡ ♥ ❡ ✳

❆ ❞ ✐ ❝ ✐ ♦ ♥ ❛ ❧ ♠ ❡ ♥ ❡ ✱ ♣ ♦ ❡ ❡ ❧ ❛ ❝ ❛ ♣ ❛ ❝ ✐ ❞ ❛ ❞ ❞ ❡ ✐ ♠ ♣ ❧ ❡ ♠ ❡ ♥ ❛ ❛ ❧ ❣ ♦ ✐ ♠ ♦ ❣ ❛ ❝ ✐ ❛ ❛ ✉ ❝ ❛ ♣ ❛ ❝ ✐ ❞ ❛ ❞ ❞ ❡  

♣ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ✱ ❧ ♦ ✉ ❡ ❤ ❛ ❝ ❡ ✉ ❡ ❡ ❛ ♠ ✉ ② ❢ ❝ ✐ ❧ ② ♣ ✐ ❞ ♦ ❡ ❧ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ♣ ♦ ❝ ❡ ♦ ❞ ❡ ❝ ❧ ❝ ✉ ❧ ♦ ✉ ❡ ❡ ♥  

♦ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❡ ❛ ♥ ❡ ❧ ❛ ✐ ✈ ❛ ♠ ❡ ♥ ❡ ❝ ♦ ♠ ♣ ❧ ❡ ❥ ♦ ♦ ❡ ① ❡ ♥ ♦ ✳

❊ ♥ ❡ ❡ ❝ ✉ ♦ ❡ ✉ ✐ ❧ ✐ ③ ❛ ❡ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ♣ ❛ ❛ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❧ ♦ ♠ ♦ ❞ ❡ ❧ ♦ ❛ ♥ ❛ ❧ ✐ ❝ ♦ ♣ ❛ ❛ ❡ ❧  

❡ ✉ ❞ ✐ ♦ ❞ ❡ ❧ ♥ ❡ ❛ ❞ ❡ ❡ ♣ ❡ ❛ ✱ ♦ ❚ ❡ ♦ ❛ ❞ ❡ ❈ ♦ ❧ ❛ ✱ ② ❛ ✉ ❡ ♣ ❡ ♠ ✐ ❡ ✉ ✐ ❧ ✐ ③ ❛ ❧ ❛ ♠ ✐ ♠ ❛ ♥ ♦ ♠ ❡ ♥ ❝ ❧ ❛ ✉ ❛ ②  

✐ ♠ ❜ ♦ ❧ ♦ ❣ ❛ ✉ ❡ ❡ ✉ ✐ ❧ ✐ ③ ❛ ❡ ♥ ❧ ❛ ♠ ❛ ② ♦ ♣ ❛ ❡ ❞ ❡ ❧ ✐ ❡ ❛ ✉ ❛ ♦ ❜ ❡ ❡ ❧ ❡ ♠ ❛ ✳

❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦ ♥ ♦ ❡ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ❛ ✉ ❡ ❝ ✉ ♠ ♣ ❧ ❛ ❡ ♥ ✉ ♦ ❛ ❧ ✐ ❞ ❛ ❞ ❝ ♦ ♥ ❧ ❛ ❝ ❛ ❛ ❝ ❡ ✐ ❝ ❛ ❞ ❡ ✉ ♥  

♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ✱ ♣ ✉ ❡ ♦ ✉ ❡ ✉ ❝ ❞ ✐ ❣ ♦ ❢ ✉ ❡ ♥ ❡ ♥ ♦ ❡ ♣ ♦ ♥ ❡ ❛ ❞ ✐ ♣ ♦ ✐ ❝ ✐ ♥ ❞ ❡ ❧ ♦ ✉ ✉ ❛ ✐ ♦ ✱ ✐ ♥ ❡ ♠ ❜ ❛ ❣ ♦ ✱  

❡ ❛ ❛ ❞ ❡ ✉ ♥ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ❣ ❛ ✉ ✐ ❛ ② ✉ ❡ ♣ ✉ ❡ ❞ ❡ ✉ ✐ ❧ ✐ ③ ❛ ❡ ♣ ❛ ❛ ❧ ♦ ♣ ♦ ♣ ✐ ♦ ✉ ❡ ❡ ❞ ❡ ❡ ❡ ✱ ♣ ♦  

❧ ♦ ✉ ❡ ❧ ❧ ❡ ♥ ❛ ❧ ♦ ❡ ✉ ❡ ✐ ♠ ✐ ❡ ♥ ♦ ♣ ❛ ❛ ❡ ✉ ✐ ❧ ✐ ③ ❛ ❞ ❛ ❡ ♥ ❡ ❡ ❝ ✉ ♦ ✳

✶ ✺ 

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✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙  

❋ ✐ ❣ ✉ ❛ ✸ ✳ ✸ ✳ ✿ ■ ♠ ❛ ❣ ❡ ♥ ❞ ❡ ❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦  

❙ ♠ ❛ ❤ ❙ ✉ ❞ ✐ ♦ ♣ ✉ ❡ ❞ ❡ ♦ ❜ ❡ ♥ ❡ ❡ ❡ ♥ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❞ ✐ ❡ ❝ ❝ ✐ ♥ ✿

❤ ♣ ✿ ✴ ✴ ❡ ♥ ✳ ♠ ❛ ❤ ✳ ✐ ♥ ❢ ♦ ✴ ❢ ♦ ✉ ♠ ✴  

✸ ✳ ✷ ✳ ✹ ✳ ❙ ❛ ❣ ❡  

❙ ❛ ❣ ❡ ❡ ✱ ✉ ✐ ③ ✱ ❡ ❧ ♣ ♦ ② ❡ ❝ ♦ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ♠ ❛ ♠ ❜ ✐ ❝ ✐ ♦ ♦ ② ❝ ♦ ♠ ♣ ❧ ❡ ♦ ❡ ① ✐ ❡ ♥ ✲  

❡ ❡ ♥ ❧ ❛ ❛ ❝ ✉ ❛ ❧ ✐ ❞ ❛ ❞ ✳ ❙ ✐ ❡ ♥ ❞ ♦ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ❛ ❡ ❧ ❛ ✐ ✈ ❛ ♠ ❡ ♥ ❡ ♥ ✉ ❡ ✈ ♦ ✱ ✐ ♥ ✐ ❝ ✐ ❛ ❞ ♦ ❡ ♥ ❡ ❧ ✷ ✵ ✵ ✺ ✱ ❤ ❛ ❧ ♦ ❣ ❛ ❞ ♦  

❝ ♦ ♥ ♦ ❧ ✐ ❞ ❛ ❡ ❡ ♥ ❧ ♦ ❝ ✐ ❝ ✉ ❧ ♦ ❛ ❝ ❛ ❞ ♠ ✐ ❝ ♦ ♠ ✉ ♥ ❞ ✐ ❛ ❧ ❡ ❝ ♦ ♠ ♦ ✉ ♥ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ♣ ❡ ❞ ✐ ❧ ❡ ❝ ❛ ♣ ♦ ✈ ❛ ✐ ❛  

❛ ③ ♦ ♥ ❡ ✳ ▲ ❛ ♣ ✐ ♠ ❡ ❛ ✱ ❙ ❛ ❣ ❡ ♥ ♦ ❡ ✐ ♥ ✈ ❡ ♥ ❧ ❛ ✉ ❡ ❞ ❛ ♣ ❛ ✐ ❡ ♥ ❞ ♦ ❞ ❡ ❝ ❡ ♦ ❡ ♥ ✉ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ✱ ✐ ♥ ♦ ✉ ❡ ❤ ❛ ❝ ❡  

✉ ♦ ❞ ❡ ♠ ✉ ❝ ❤ ❛ ♦ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ② ❛ ❡ ① ✐ ❡ ♥ ❡ ❧ ❛ ❝ ✉ ❛ ❧ ❡ ✐ ♥ ❡ ❣ ❛ ❡ ♥  

✉ ♥ ❛ ♦ ❧ ❛ ✐ ♥ ❡ ❢ ❛ ③ ✱ ❧ ♦ ✉ ❡ ❧ ♦ ❤ ❛ ❝ ❡ ❡ ① ❡ ♠ ❛ ❞ ❛ ♠ ❡ ♥ ❡ ♣ ♦ ❞ ❡ ♦ ♦ ② ✈ ❡ ✐ ❧ ✳ ❖ ❛ ❛ ③ ♥ ❡ ❡ ❧ ❤ ❡ ❝ ❤ ♦ ✉ ❡  

❝ ♦ ♠ ❜ ✐ ♥ ❛ ♦ ❞ ❛ ❧ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❝ ♦ ♥ ❡ ❧ ♣ ♦ ❞ ❡ ② ❡ ♥ ❝ ✐ ❧ ❧ ❡ ③ ❞ ❡ ❧ ❧ ❡ ♥ ❣ ✉ ❛ ❥ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ❞ ❡ ♠  

♣ ✐ ❞ ♦ ❝ ❡ ❝ ✐ ♠ ✐ ❡ ♥ ♦ ✿ ② ❤ ♦ ♥ ✳

▲ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ❙ ❛ ❣ ❡ ❛ ❜ ❛ ❝ ❛ ♥ ✉ ♥ ❛ ❛ ♠ ♣ ❧ ✐ ❛ ❣ ❛ ♠ ❛ ❞ ❡ ❡ ♠ ❛ ✱ ❛ ❧ ❡ ❝ ♦ ♠ ♦ ❧ ❣ ❡ ❜ ❛ ✱ ❝ ❧ ❝ ✉ ❧ ♦ ✱  

❛ ❧ ❣ ❡ ❜ ❛ ❧ ✐ ♥ ❡ ❛ ❧ ✱ ❡ ♦ ❛ ❞ ❡ ♥ ♠ ❡ ♦ ✱ ❡ ♦ ❛ ❞ ❡ ❣ ❛ ❢ ♦ ✱ ❡ ❝ ✳ ❊ ♥ ❧ ♦ ✉ ❡ ❡ ♣ ❡ ❝ ❛ ❛ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✱ ❙ ❛ ❣ ❡  

♣ ✉ ❡ ❞ ❡ ❤ ❛ ❝ ❡ ♦ ❞ ♦ ❧ ♦ ✉ ❡ ▼ ❛ ① ✐ ♠ ❛ ❤ ❛ ❝ ❡ ✱ ♣ ✉ ❡ ♦ ✉ ❡ ▼ ❛ ① ✐ ♠ ❛ ❡ ✉ ♥ ♦ ❞ ❡ ❧ ♦ ❝ ♦ ♠ ♣ ♦ ♥ ❡ ♥ ❡ ❞ ❡ ❙ ❛ ❣ ❡ ✳

❊ ♥ ❛ ❞ ✐ ❝ ✐ ♥ ✱ ❙ ❛ ❣ ❡ ♣ ♦ ❡ ❡ ❛ ♠ ♣ ❧ ✐ ❛ ❝ ❛ ♣ ❛ ❝ ✐ ❞ ❛ ❞ ❡ ❞ ❡ ❛ ❜ ❛ ❥ ❛ ❝ ♦ ♥ ❣ ❛ ❢ ♦ ✱ ❧ ♦ ✉ ❡ ❧ ♦ ❤ ❛ ❝ ❡ ✐ ❞ ❡ ❛ ❧ ♣ ❛ ❛ ❧ ❛  

❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❞ ❡ ❛ ❧ ❣ ♦ ✐ ♠ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ❞ ❡ ❡ ❞ ❡ ✱ ❝ ♦ ♠ ♦ ❧ ❛ ✉ ❛ ♠ ❝ ♦ ❛ ② ❡ ❧ ❜ ♦ ❧ ♠ ♥ ✐ ♠ ♦ ❞ ❡  

❡ ① ♣ ❛ ♥ ✐ ♥ ✱ ♣ ♦ ♠ ❡ ♥ ❝ ✐ ♦ ♥ ❛ ❛ ❧ ❣ ✉ ♥ ♦ ✳

❊ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❞ ❡ ❧ ♣ ♦ ② ❡ ❝ ♦ ❙ ❛ ❣ ❡ ❡ ♠ ✉ ② ❝ ❧ ❛ ♦ ✱ ❝ ♦ ♥ ✈ ❡ ✐ ❡ ❡ ♥ ✉ ♥ ❛ ❛ ❧ ❡ ♥ ❛ ✐ ✈ ❛ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡  

♣ ❛ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❝ ♦ ♠ ❡ ❝ ✐ ❛ ❧ ❡ ❝ ♦ ♠ ♦ ▼ ❛ ❧ ❛ ❜ ✱ ▼ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ✱ ▼ ❛ ♣ ❧ ❡ ✱ ❡ ❝ ✳ ♦ ❧ ♦ ✉ ❡ ❡ ♣ ✉ ❡ ❞ ❡ ✈ ❡ ✱ ❡ ♥  

♣ ♦ ❝ ♦ ✐ ❡ ♠ ♣ ♦ ❡ ❤ ❛ ❛ ✈ ❛ ♥ ③ ❛ ❞ ♦ ✉ ♥ ❣ ❛ ♥ ❡ ❝ ❤ ♦ ❤ ❛ ❝ ✐ ❛ ❡ ❧ ❧ ♦ ❣ ♦ ❞ ❡ ❛ ❧ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ✳ ❊ ♥ ♠ ✐ ♦ ♣ ✐ ♥ ✐ ♥ ✱ ❡ ♥ ✉ ♥ ♦  

❝ ✉ ❛ ♥ ♦ ❛ ♦ ✱ ❙ ❛ ❣ ❡ ❡ ❧ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ❞ ❡ ✉ ♦ ♠ ❛ ❣ ❡ ♥ ❡ ❛ ❧ ✐ ③ ❛ ❞ ♦ ❡ ♥ ❡ ❧ ❛ ❝ ♦ ♠ ✉ ♥ ✐ ❞ ❛ ❞ ❝ ✐ ❡ ♥ ✜ ❝ ❛ ②  

❛ ❝ ❛ ❞ ♠ ✐ ❝ ❛ ❡ ♥ ❡ ❧ ♠ ✉ ♥ ❞ ♦ ✳

❙ ❛ ❣ ❡ ♣ ✉ ❡ ❞ ❡ ❡ ♦ ❜ ❡ ♥ ✐ ❞ ♦ ❡ ♥ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❞ ✐ ❡ ❝ ❝ ✐ ♥ ✿

❤ ♣ ✿ ✴ ✴ ❛ ❣ ❡ ♠ ❛ ❤ ✳ ♦ ❣ ✴  

✸ ✳ ✷ ✳ ✺ ✳ ❖ ❝ ❛ ✈ ❡  

❖ ❝ ❛ ✈ ❡ ❡ ✉ ♥ ❛ ❤ ❡ ❛ ♠ ✐ ❡ ♥ ❛ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ❝ ❧ ❛ ✐ ✜ ❝ ❛ ❡ ❞ ❡ ♥ ♦ ❞ ❡ ❧ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ❈ ❧ ❝ ✉ ❧ ♦ ◆ ✉ ♠ ✲  

✐ ❝ ♦ ✳ ❙ ✉ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ❡ ❝ ❡ ❛ ✉ ♥ ❛ ❛ ❧ ❡ ♥ ❛ ✐ ✈ ❛ ❞ ❡ ♦ ❢ ✇ ❛ ❡ ❧ ✐ ❜ ❡ ♣ ❛ ❛ ▼ ❛ ❧ ❛ ❜ ② ❞ ❡ ❤ ❡ ❝ ❤ ♦ ✱ ❛ ❝ ✉ ❛ ❧ ♠ ❡ ♥ ❡  

❡ ① ✐ ❡ ✉ ♥ ❛ ❧ ♦ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ❝ ♦ ♠ ♣ ❛ ✐ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ❡ ♥ ❡ ❛ ♠ ❜ ❛ ✱ ❝ ❡ ❝ ❛ ♥ ❛ ❛ ❧ ✾ ✾ ✪ ✱ ❡ ❞ ❡ ❝ ✐ ✉ ❡ ❡ ❧ ❝ ❞ ✐ ❣ ♦ ❞ ❡  

▼ ❛ ❧ ❛ ❜ ♣ ✉ ❡ ❞ ❡ ❡ ❥ ❡ ❝ ✉ ❛ ❡ ❝ ❛ ✐ ✐ ♥ ♠ ♦ ❞ ✐ ✜ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❡ ♥ ❖ ❝ ❛ ✈ ❡ ✳ ❙ ✉ ❢ ♦ ❛ ❧ ❡ ③ ❛ ❛ ❞ ✐ ❝ ❛ ♥ ❡ ♥ ❧ ❛ ❡ ♥ ♦ ♠ ❡  

❝ ❛ ♣ ❛ ❝ ✐ ❞ ❛ ❞ ❞ ❡ ♣ ♦ ❝ ❡ ❛ ❣ ❛ ♥ ❞ ❡ ❝ ❛ ♥ ✐ ❞ ❛ ❞ ❡ ❞ ❡ ❞ ❛ ♦ ❛ ✉ ♥ ❛ ❣ ♥ ✈ ❡ ❧ ♦ ❝ ✐ ❞ ❛ ❞ ❞ ❡ ❝ ❧ ❝ ✉ ❧ ♦ ✱ ❛ ❝ ♦ ♠ ♦  

✶ ✻ 

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✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙  

❋ ✐ ❣ ✉ ❛ ✸ ✳ ✹ ✳ ✿ ■ ♠ ❛ ❣ ❡ ♥ ❡ ❞ ❡ ❙ ❛ ❣ ❡  

✶ ✼ 

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✸ ✳ ❍ ❊ ❘ ❘ ❆ ▼ ■ ❊ ◆ ❚ ❆ ❙ ■ ◆ ❋ ❖ ❘ ▼ ➪ ❚ ■ ❈ ❆ ❙  

❋ ✐ ❣ ✉ ❛ ✸ ✳ ✺ ✳ ✿ ■ ♠ ❛ ❣ ❡ ♥ ❡ ❞ ❡ ❖ ❝ ❛ ✈ ❡  

❡ ❧ ♠ ❛ ♥ ❡ ❥ ♦ ❡ ♥ ❢ ♦ ♠ ❛ ♥ ❛ ✉ ❛ ❧ ❞ ❡ ❡ ✉ ❝ ✉ ❛ ♠ ❛ ✐ ❝ ✐ ❛ ❧ ❡ ✱ ❧ ♦ ❝ ✉ ❛ ❧ ❧ ❛ ❤ ❛ ❝ ❡ ✐ ❞ ❡ ❛ ❧ ♣ ❛ ❛ ❛ ❜ ❛ ❥ ❛ ❝ ♦ ♥  

❛ ❧ ❣ ♦ ✐ ♠ ♦ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✱ ♣ ✉ ❡ ♦ ✉ ❡ ♠ ✉ ❝ ❤ ♦ ❞ ❡ ♦ ❡ ❜ ❛ ❛ ♥ ❡ ♥ ❝ ❧ ❝ ✉ ❧ ♦ ♠ ❛ ✐ ❝ ✐ ❛ ❧ ✳

▼ ❛ ✉ ❡ ✉ ♥ ❛ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥ ✱ ❛ ♥ ♦ ▼ ❛ ❧ ❛ ❜ ❝ ♦ ♠ ♦ ❖ ❝ ❛ ✈ ❡ ♦ ♥ ♠ ❛ ❜ ✐ ❡ ♥ ✉ ♥ ❧ ❡ ♥ ❣ ✉ ❛ ❥ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥  

② ✉ ♥ ❡ ♥ ♦ ♥ ♦ ❞ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ♣ ❛ ❛ ❡ ❧ ❝ ❧ ❝ ✉ ❧ ♦ ❝ ✐ ❡ ♥ ✜ ❝ ♦ ✳ ❊ ① ✐ ❡ ✉ ♥ ❛ ❣ ♥ ❝ ❛ ♥ ✐ ❞ ❛ ❞ ❞ ❡  

♣ ❛ ✉ ❡ ❡ ❛ ❞ ✐ ❝ ✐ ♦ ♥ ❛ ❧ ❡ ✉ ❡ ♣ ✉ ❡ ❞ ❡ ♥ ❞ ❡ ❝ ❛ ❣ ❛ ❡ ② ✉ ❡ ❡ ① ✐ ❡ ♥ ❞ ❡ ♥ ❧ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❧ ✐ ❞ ❛ ❞ ❡ ❞ ❡ ❖ ❝ ❛ ✈ ❡ ❡ ♥  

❡ ❛ ❡ ♣ ❡ ❝ ✜ ❝ ❛ ✳ ❊ ① ✐ ❡ ♥ ♣ ❛ ✉ ❡ ❡ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ♣ ❛ ❛ ❡ ❧ ❡ ❛ ❞ ❡ ♦ ♣ ✐ ♠ ✐ ③ ❛ ❝ ✐ ♥ ✱ ❛ ♥ ♦ ❧ ✐ ♥ ❡ ❛ ❧ ❝ ♦ ♠ ♦ ♥ ♦  

❧ ✐ ♥ ❡ ❛ ❧ ✳

◗ ✉ ✐ ③ ✉ ♥ ❛ ❞ ❡ ❧ ❛ ♠ ❛ ② ♦ ❡ ❞ ❡ ✈ ❡ ♥ ❛ ❥ ❛ ❞ ❡ ❖ ❝ ❛ ✈ ❡ ❡ ❧ ❛ ❢ ❛ ❧ ❛ ❞ ❡ ✉ ♥ ❛  ■ ♥ ❡ ❢ ❛ ❡ ● ❢ ✐ ❝ ❛ ❞ ❡  

❯ ✉ ❛ ✐ ♦  

♦ ● ❯ ■ ♣ ♦ ✉ ✐ ❣ ❧ ❛ ❡ ♥ ✐ ♥ ❣ ❧ ✳ ❆ ✉ ♥ ✉ ❡ ❡ ① ✐ ❡ ♥ ❛ ❧ ❣ ✉ ♥ ♦ ♣ ♦ ② ❡ ❝ ♦ ✉ ❡ ❡ ❤ ❛ ♥ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❞ ♦  

❡ ♥ ❢ ♦ ♠ ❛ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ✐ ❡ ♥ ❡ ♣ ❛ ❛ ✉ ❜ ❛ ♥ ❛ ❡ ❛ ❞ ❡ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ✱ ❛ ♥ ♥ ♦ ❡ ① ✐ ❡ ✉ ♥ ❛ ✐ ♥ ❡ ❢ ❛ ③ ✉ ❡ ❧ ❧ ❡ ♥ ❡ ❡ ♥ ✉  

♦ ❛ ❧ ✐ ❞ ❛ ❞ ❡ ❡ ✈ ❛ ❝ ♦ ✱ ❛ ✉ ♥ ✉ ❡ ② ❛ ❡ ❤ ❛ ❛ ♥ ✉ ♥ ❝ ✐ ❛ ❞ ♦ ♣ ♦ ♣ ❛ ❡ ❞ ❡ ❧ ♦ ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❞ ♦ ❡ ❞ ❡ ❧ ♣ ♦ ② ❡ ❝ ♦  

❖ ❝ ❛ ✈ ❡ ✉ ❡ ② ❛ ❡ ❡ ❛ ❜ ❛ ❥ ❛ ♥ ❞ ♦ ❡ ♥ ✉ ♥ ❛ ● ❯ ■ ✉ ❡ ✈ ❡ ♥ ❞ ✐ ♥ ❡ ❣ ❛ ❞ ❛ ❡ ♥ ❢ ✉ ✉ ❛ ✈ ❡ ✐ ♦ ♥ ❡ ✳

❖ ❝ ❛ ✈ ❡ ♣ ✉ ❡ ❞ ❡ ❞ ❡ ❝ ❛ ❣ ❛ ❡ ❞ ❡ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❞ ✐ ❡ ❝ ❝ ✐ ♥ ✿

❤ ♣ ✿ ✴ ✴ ✇ ✇ ✇ ✳ ❣ ♥ ✉ ✳ ♦ ❣ ✴ ♦ ❢ ✇ ❛ ❡ ✴ ♦ ❝ ❛ ✈ ❡ ✴  

✸ ✳ ✷ ✳ ✻ ✳ ❙ ❝ ✐ ❧ ❛ ❜  

❙ ❝ ✐ ❧ ❛ ❜ ❡ ✉ ♥ ♦ ❢ ✇ ❛ ❡ ♠ ✉ ② ✐ ♠ ✐ ❧ ❛ ❛ ❖ ❝ ❛ ✈ ❡ ② ✉ ❡ ❛ ♠ ❜ ✐ ♥ ♣ ❡ ❡ ♥ ❡ ❝ ❡ ❛ ❧ ❛ ❝ ❛ ❡ ❣ ♦ ❛ ❞ ❡ ♦ ❢ ✇ ❛ ❡  

♣ ❛ ❛ ❝ ❧ ❝ ✉ ❧ ♦ ♥ ✉ ♠ ✐ ❝ ♦ ✳ ❆ ✉ ♥ ✉ ❡ ✉ ❝ ♦ ♠ ♣ ❛ ✐ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ❝ ♦ ♥ ▼ ❛ ❧ ❛ ❜ ♥ ♦ ❡ ❛ ♥ ❛ ❧ ❛ ❝ ♦ ♠ ♦ ❖ ❝ ❛ ✈ ❡ ✱ ♣ ✉ ❡ ❞ ❡  

❡ ❝ ♦ ♥ ✐ ❞ ❡ ❛ ❞ ♦ ✉ ♥ ❛ ❛ ❧ ❡ ♥ ❛ ✐ ✈ ❛ ✳ ❆ ❧ ✐ ❣ ✉ ❛ ❧ ✉ ❡ ❖ ❝ ❛ ✈ ❡ ✱ ❙ �