In this paper we propose new method for proving of global solutions for 3D Navier-Stokes equations. This complies an application to the Clay Institute Millennium Prize Navier Stokes Problem. The proposed method can be applied for investigation of global solutions for other classes of PDEs.
We construct a function for almost-complex Riemannian manifolds. Non-vanishing of the function for the almost-complex structure implies the almost-complex structure is not integrable. Therefore the constructed function is an obstruction for the existence of complex structures from the almost-complex structure. It is a function, not a tensor, so it is easier to work with.
This paper proves that there does not exist a polynomial-time algorithm to the the subset sum problem. As this problem is in NP, the result implies that the class P of problems admitting polynomial-time algorithms does not equal the class NP of problems admitting nondeterministic polynomial-time algorithms.
Some properties of the optimal representation of numbers are investigated. This representation, which is to the base-e, is examined for coding of integers. An approximate representation without fractions that we call WF is introduced and compared with base-2 and base-3 representations, which are next to base-e in efficiency. Since trees are analogous to number representation, we explore the relevance of the statistical optimality of the base-e system for the understanding of complex system behavior and of social networks. We show that this provides a new theoretical explanation for the nature of the power law exhibited by many open complex systems. In specific, we show that the power law distribution most often proposed for such systems has a form that is similar to that derived from the optimal base-e representation.