We prove that P ≠ NP by proving the existence of a class of functions we call Τ (Greek Tau), each of whose members satisfies the conditions of one-way functions. Each member of Τ is a function computable in polynomial time, with negligible probability of finding its inverse by any polynomial probabilistic algorithm. We also prove that no polynomial-time algorithm exists to compute the inverse o...

We introduce a method and algorithm for computing the weighted MoorePenrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [24] to multiple variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Al...