The Solutions of the 3rd and 4th Millennium Mathematical Problems

In this book are described the solutions of the 3rd and 4th Millennium problems of the Clay Mathematical Institute. The first problem is in computer science and in particular in computational complexity, and is known as the P (polynomial time) versus NP (non-deterministic polynomial time) problem. The solution is that the NP complexity class is equivalent to the exponential time complexity class. Thus the standard passwords setting is safe. The second problem is in mathematical fluid dynamics of 3-dimensional incompressible flows described by the Navier-Stokes equations. The solution is the expected that the unique smooth local solution under appropriate initial conditions, that include finite initial energy, can be extended smoothly for all times and there is no spontaneous blow-up in the flow, within finite time.