This paper considers a new canonical duality theory for solving mixed
integer quadratic programming problem. It shows that this well-known NP-hard
problem can be converted into concave maximization dual problems without duality
gap. And the dual problems can be solved, under certain conditions, by polynomial algorithms.
When we study Lorentz transformation in the framework of quantum gauge
theory of gravity, we will find that the vacuum gravitational gauge field will
be changed under gravitational gauge transformation, which will change the
structure of the physical space-time and cause clock dilation effect. The study
in this paper provides us with new insights to understand the essential and
intrinsic relation between special relativity and general relativity. It
provides us with a new way to unify special relativity and general relativity.
A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.