Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Nuclear Pore Complex Number and Distribution throughout the Saccharomyces cerevisiae Cell Cycle by Three-Dimensional Reconstruction from Electron Micrographs of Nuclear Envelopes

The number of nuclear pore complexes (NPCs) in individual nuclei of the yeast Saccharomyces cerevisiae was determined by computer-aided reconstruction of entire nuclei from electron micrographs of serially sectioned cells. Nuclei of 32 haploid cells at various points in the cell cycle were modeled and found to contain between 65 and 182 NPCs. Morphological markers, such as cell shape and nuclear shape, were used to determine the cell cycle stage of the cell being examined. NPC number was correlated with cell cycle stage to reveal that the number of NPCs increases steadily, beginning in G1-phase, suggesting that NPC assembly occurs continuously throughout the cell cycle. However, the accumulation of nuclear envelope observed during the cell cycle, indicated by nuclear surface area, is not continuous at the same rate, such that the density of NPCs per unit area of nuclear envelope peaks in apparent S-phase cells. Analysis of the nuclear envelope reconstructions also revealed no preferred NPC-to-NPC distance. However, NPCs were found in large clusters over regions of the nuclear envelope. Interestingly, clusters of NPCs were most pronounced in early mitotic nuclei and were found to be associated with the spindle pole bodies, but the functional significance of this association is unknown.