Commit e9d1e9cd by Michael Keller

### .

parent aeb6b0a5
 ... ... @@ -213,8 +213,18 @@ statement with a condition on $R$ might be helpful: &+ R(c_\beta, c_\alpha) \cdot ((u_\beta + \lambda) \cdot (v_\alpha + \lambda) - u_\beta \cdot v_{\alpha})\\ &+ R(c_\beta, c_\beta) \cdot ((u_\beta + \lambda) \cdot (v_\beta - \lambda) - u_\beta \cdot v_{\beta})\\ &= R(c_\alpha, c_\alpha) \cdot (u_\alpha \cdot \lambda - v_\alpha \cdot \lambda - \lambda^2)\\ &+ R(c_\alpha, c_\beta) \cdot (- u_\alpha \cdot \lambda - v_\beta \cdot \lambda + \lambda^2)\\ &- R(c_\alpha, c_\beta) \cdot (u_\alpha \cdot \lambda + v_\beta \cdot \lambda - \lambda^2)\\ &+ R(c_\beta, c_\alpha) \cdot (u_\beta \cdot \lambda + v_\alpha \cdot \lambda + \lambda^2)\\ &+ R(c_\beta, c_\beta) \cdot (- u_\beta \cdot \lambda + v_\beta \cdot \lambda - \lambda^2)\\ &- R(c_\beta, c_\beta) \cdot (u_\beta \cdot \lambda - v_\beta \cdot \lambda + \lambda^2)\\ &= \lambda^2 \cdot \left ( R(c_\alpha, c_\beta) + R(c_\beta, c_\alpha) - R(c_\alpha, c_\alpha) - R(c_\beta, c_\beta) \right )\\ &+ \lambda \cdot (R(c_\alpha, c_\alpha) - R(c_\alpha, c_\beta)) \cdot u_\alpha\\ &+ \lambda \cdot (R(c_\beta, c_\alpha) - R(c_\beta, c_\beta)) \cdot u_\beta\\ &+ \lambda \cdot (R(c_\beta, c_\alpha) - R(c_\alpha, c_\alpha)) \cdot v_\alpha\\ &+ \lambda \cdot (R(c_\beta, c_\beta) - R(c_\alpha, c_\beta)) \cdot v_\beta \end{align*} Which, if every crop likes all other crops more than itself, can be made positive with the appropriate $\lambda$. Then the quadratic term in the formula above will always be positive, and we can focus only on all of the linear constraints. \end{proof} \ No newline at end of file
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