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alleles assume asymptotic solutions Banach spaces boundary condition chromosome circular symmetry coefficients completely continuous operator consider constant converges Corollary corresponding defined denote derived differential equation differential inequality domain eigenvalues ellipsoidal wave functions example existence theorems expansion finite follows formal solution formula gametes given Hence Hermitian Hilbert space homologous genes implicit function theorem implies inbreeding individuals inequality integral equation interval inverse function theorem Laplace transform least characteristic value Lemma linear operator Math Mathematics Research Center matrix method monoecious MRC Technical Summary neighborhood non-singular obtain periodic solution piecewise-linear function power series probability proof of Theorem properties prove random mating rearrangement respectively result satisfied scalar solvable Suppose symmetrically decreasing Technical Summary Report Theorem 4.1 two-body problem unique solution UNITED STATES ARMY variance effective number vector zero