NEW PROGRESS IN STATIONARY D=N=4 SUPERGRAVITY

A bosonic sector of the four-dimensional low-energy heterotic string theory with two Abelian gauge fields is considered in the stationary case. A new 4× 4 unitary null-curvature matrix representation of the theory is derived and the corresponding formulation based on the use of a new 2 × 2 Ernst type matrix complex potential is developed. The group of hidden symmetries is described and classified in the matrix-valued quasi general relativity form. A subgroup of charge symmetries is constructed and representation which transforms linearly under the action of this symmetry subgroup is established. Also the solution generation procedure based on the application of the total charge symmetry subgroup to the stationary Einstein–Maxwell theory is analyzed.

A Geometric Theory for Formulation of Form, Profile and Orientation Tolerances: Problem Formulation

This paper develops a geometric theory which unifies the formulation and evaluation of form (straightness, flatness, cylindricity and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. In the paper, based on an an important observation that a toleranced feature exhibits a symmetry subgroup G0 under the action of the Euclidean group, SE(3), we identify the configuration space of a toleranced (or a symmetric) feature with the homogeneous space SE(3)/G0 of the Euclidean group. Geometric properties of SE(3)/G0, especially its exponential coordinates carried over from that of SE(3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE(3)/G0, with G0 being the symmetry subgroup of the underlying feature. We transform the non-differentiable minimization problem into a differentiable minimization problem over an extended configuration space. Using geometric properties of SE(3)/G0, we derive a sequence of linear programming problems whose solutions can be used to approximate the minimum zone solutions.