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Home » استخدم نظرية فيثاغورس: \(10^2 = 8^2 + \textالقاعدة^2\). - Crosslake

استخدم نظرية فيثاغورس: \(10^2 = 8^2 + \textالقاعدة^2\). - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 This is the general relationship. Since the conditions determine \( a = 1 \), but the problem asks for the relationship, we state it as:

📌 Thus, the required relationship is \(\boxed{c = 9a - 5}\).

📌 In a neural network optimization, the loss function is modeled as \( L(w) = w^2 - 2mw + m^2 + 4 \), where \( w \) is a weight and \( m \) is a hyperparameter. Find the value of \( m \) such that the minimum loss is exactly 1.

📌 The function \( L(w) = w^2 - 2mw + m^2 + 4 \) is a quadratic in \( w \). The minimum occurs at \( w = \frac{2m}{2} = m \), since the vertex of \( aw^2 + bw + c \) is at \( w = -b/(2a) \), here \( a = 1 \), \( b = -2m \), so:

📌 w = \frac{2m}{2} = m

📌 Substitute \( w = m \) into \( L(w) \):

📌 L(m) = (m)^2 - 2m(m) + m^2 + 4 = m^2 - 2m^2 + m^2 + 4 = 0 + 4 = 4

📌 But the problem states the minimum loss must be 1, yet we get 4. This suggests either the function is misstated or we must adjust. However, observe:

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