Lines Matching refs:solution

66 interested positive solution $x_t$. Again, if there are multiple $x_t$ solutions, we may want to
72 **Theorem 1.** The solution to $x_t$ is
79 3 may have no solution at all if $(r_1^2 - 1) y^2 + r_1^2 x^2 < 0$.
177 Therefore $||P C|| = ||P_1 C_1|| \cdot (||C_f C|| / ||C_f C_1||) = r_1 x'$. Thus $x'$ is a solution
181 **Lemma 2.** For every solution $x_t$, if we extend/shrink segment $C_f P$ to $C_f P_1$ with ratio
191 * when $r_1 > 1$, there's always one unique intersection/solution; we call this "well-behaved"; this
193 * when $r_1 = 1$, there's either one or zero intersection/solution (excluding $C_f$ which is always
206 **Lemma 3.** When solution exists, one such solution is
263 **Corollary 2.** If $r_1 = 1$, then the solution $x_t = \frac{x^2 + y^2}{(1 + r_1) x}$, and
268 **Corollary 3.** If $r_1 > 1$, then the unique solution is
295 denomenator) is always valid because $r_1 > 1$ and it's the unique solution due to Corollary 1.
300 1. there's no solution to $x_t$ if $(r_1^2 - 1) y^2 + r_1^2 x^2 < 0$
306 (Note that solution $x_t$ still has to be nonnegative to be valid; also note that
307 $x_t > 0 \Leftrightarrow x > 0$ if the solution exists.)
318 By analysis similar to Lemma 3., the solution to $x_t$ does not depend on the sign of $x$ and