Equation (9) has three solutions, and the optimal solution is given by {boldsymbol{w}}_{boldsymbol{o}}={{tilde{boldsymbol{R}}}_{s^2}}^{-1}{tilde{boldsymbol{P}}}_{d,s}.
First note the trigonometric identity begin{aligned} sum_{m=1}^{q}e biggl( frac{nm}{q} biggr) =textstylebegin{cases} q &mbox{if }qmid n, 0 &mbox{if }qnmid n. end{cases}displaystyle end{aligned} (5) Since (pequiv 1operatorname{mod} 3), the congruence equation (x^{3}equiv 1 operatorname{mod} p) has three solutions.
Moreover, one of the following cases occurs: (i) is even and (1.1) has two sign-changing solutions, (ii) is even and (1.1) has six solutions, three of which are of the same sign, (iii) is odd and (1.1) has two sigh-changing solutions, (iv) is odd and (1.1) has three solutions of the same sign.
. is even and (1.1) has two sign-changing solutions, is even and (1.1) has six solutions, three of which are of the same sign, is odd and (1.1) has two sigh-changing solutions, is odd and (1.1) has three solutions of the same sign.
Therefore, problem (7) has three solutions u 1, u 2 and u 3 with u 1 ∈ Ω α 1 β 1, u 2 ∈ Ω α 2 β 2 and u 3 ∈ Ω α 1 β 2 ∖ Ω ¯ α 1 β 1 ∪ Ω ¯ α 2 β 2. By the facts that all solutions of (7) satisfy [ α 1, β 2 ] and are solution of (1), the proof is complete.
In the stationary state in which the left-hand side of (4) vanishes, if the mean c 1 * = z o, the allele frequency p i * has three solutions, namely 1 / 2 and (1 ± 1 − (γ ^ / γ i ) 2 ) / 2, where γ ^ = 2 2 μ / s.
