The crossover operation is applied to each pair of the generated mutant vector and its corresponding parent vector using the following equations:

這句話中的using the following equations做是方式狀語嗎，為什么using前不加by？

using the following equations 是現在分詞短語，作方式狀語。

by using the following equations 表達同樣的意義，但這是介詞短語，作方式狀語。

二者可以交換運用。比較而言，前者更簡練。