## Algebraic, Geometric, and Stochastic Aspects of Genetic Operators, Volume 2099Genetic algorithms for function optimization employ genetic operators patterned after those observed in search strategies employed in natural adaptation. Two of these operators, crossover and inversion, are interpreted in terms of their algebraic and geometric properties. Stochastic models of the operators are developed which are employed in Monte Carlo simulations of their behavior. |

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adaptive plan algebraic allele associated permutation basis and reflection basis set blfj(b bound is attainable bounding sphere centroid chromosome Computer coordinate value coordinate-bounded Corollary 2.2 corresponding crossover and inversion crossover operators defined Definition denote density functions distance of points easily verified Euclidean space evaluate Expectation following Theorem 2.3 function composition function space game configurations game trees genetic algorithms genetic operators goal point heuristics hypersphere i=l i=l iff i e a inner product intuitive inversion operators inversion pattern isomorphic itn coordinate Kl K2 Lemma locus Markov Chain matrix maximal M.B.S. maximal radius metric metric space minimal bounding Monte Carlo simulations n-sphere centered Notation obvious penalty function plane polytope point in VSTR-space Proof quadrant random variables Remark result Section set of points strategies string Suppose Theorem 2.4 vector volume-uniform distribution VSTR x(t+l yield