Reconfiguration Techniques and Geometric Constraints of Metamorphic Mechanisms

This paper proposes a geometric way to generate metamorphic configurations and investigates metamorphic principles based on geometrized displacement group. Metamorphic reconfiguration techniques are revealed as the variations of kinematic joints, kinematic links and geometric orientation constraints particularly by examining the invariant configuration properties of a mechanism. The nature of all these configuration changes belongs to geometric constraint category. Metamorphic configuration units are proposed as the irreducible reconfiguration modules to envelop these reconfiguration techniques. It can self-reconfigure or be combined to generate metamorphosis. Moreover, the geometrized displacement group is lent to achieve a geometric representation for configuration modelling and further reconfiguration operations. Based on seting up kinematic group extended qualitatively according to its group structure, geometrized displacement group modelling is proposed for these identified metamorphic configuration units. The investigated group motion-matrix is an integration of its displacement group properties and kinematic extensions. Then defined geometric constraint relations and the proposed dependence rules lead to metamorphic principles. In this way, metamorphic process is mapped to matrix operations under group extensions and their compositions. Design examples and a metamorphic joint with six configurations are given to illustrate the feasibility of these metamorphic principles.

AN EFFICIENT DENSITY FUNCTIONAL ALGORITHM IN A STRONG MAGNETIC FIELD

We describe the implementation of a manifestly gauge invariant configuration space method for ab initio electronic structure calculations in an arbitrarily strong external magnetic field. To be able to reproduce empirical data for realistic systems, we will also formulate our real-space algorithm in magnetic fields for non-local ionic pseudo-potentials. Numerical applications focus on two issues, namely a careful assessment of the convergence properties of our algorithm and in particular the implications of our gauge invariant formulation, and the calculation of NMR shifts for a number of typical molecules.

The even master system and generalized Kummer surfaces

In this paper we study a generalized Kummer surface associated to the Jacobian of those complex algebraic curves of genus two which admit an automorphism of order three. Such a curve can always be written as y2 = x6 + 2kx3 + 1 and k2 ╪ 1 is the modular parameter. The automorphism extends linearly to an automorphism of the Jacobian and we show that this extension has a 94 invariant configuration, i.e. it has 9 fixed points and 9 invariant theta curves, each of these curves contains 4 fixed points and 4 invariant curves pass through each fixed point. The quotient of the Jacobian by this automorphism has 9 singular points of type A2 and the 94 configuration descends to a 94 configuration of points and lines, reminiscent to the well-known 166 configuration on the Kummer surface. Our ‘generalized Kummer surface’ embeds in ℙ4 and is a complete intersection of a quadric and a cubic hypersurface. Equations for these hypersurfaces are computed and take a very symmetric form in well-chosen coordinates. This computation is done by using an integrable system, the ‘even master system’.