Recently, Long proposed a new type of quantum computers called duality computers or duality quantum computers. The duality computers based on the general quantum interference principle are much more powerful than an ordinary quantum computer. A mathematical theory for the duality computers has been presented by Gudder. However, he pointed out that a paradoxical situation of the mathematical theory occurs between the mixed state formalism and the pure state formalism. This paper argues for Gudder's mathematical theory of the duality computers for the mixed state formalism. First, we point out two problems existing in the pure state description of the duality computers. Then, we present a new mathematical theory of the duality computers for the pure state formalism according with Gudder's mixed state description, generalize the new mathematical theory of the duality computers in the density matrix formalism, and discuss some basic properties of the divider operators and combiner operators of the duality computers. The new mathematical theory can conquer the two problems mentioned above. Finally, we find that the nonunitary operations can be performed on every path of a quantum wave divider of the duality computers. Especially, we discuss in detail that the subwaves interact with environment by a CNOT gate.