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.

Lessons from the past: Evolutionary impacts of mass extinctions

Mass extinctions have played many evolutionary roles, involving differential survivorship or selectivity of taxa and traits, the disruption or preservation of evolutionary trends and ecosystem organization, and the promotion of taxonomic and morphological diversifications—often along unexpected trajectories—after the destruction or marginalization of once-dominant clades. The fossil record suggests that survivorship during mass extinctions is not strictly random, but it often fails to coincide with factors promoting survival during times of low extinction intensity. Although of very serious concern, present-day extinctions have not yet achieved the intensities seen in the Big Five mass extinctions of the geologic past, which each removed ≥50% of the subset of relatively abundant marine invertebrate genera. The best comparisons for predictive purposes therefore will involve factors such as differential extinction intensities among regions, clades, and functional groups, rules governing postextinction biotic interchanges and evolutionary dynamics, and analyses of the factors that cause taxa and evolutionary trends to continue unabated, to suffer setbacks but resume along the same trajectory, to survive only to fall into a marginal role or disappear (“dead clade walking”), or to undergo a burst of diversification. These issues need to be addressed in a spatially explicit framework, because the fossil record suggests regional differences in postextinction diversification dynamics and biotic interchanges. Postextinction diversifications lag far behind the initial taxonomic and morphological impoverishment and homogenization; they do not simply reoccupy vacated adaptive peaks, but explore opportunities as opened and constrained by intrinsic biotic factors and the ecological and evolutionary context of the radiation.

The law of mass action governs antigen-stimulated cytolytic activity of CD8+ cytotoxic T lymphocytes.

An analysis of the initial antigen-recognition step in the destruction of target cells by CD8+ cytolytic T lymphocytes (CTLs) shows that a relationship in the form of the law of mass action can be used to describe interactions between antigen-specific receptors on T cells (TCRs) and their natural ligands on target cells (peptide-major histocompatibility protein complexes, termed pepMHC complexes), even though these reactants are confined to their respective cell membranes. For a designated level of lysis and receptor affinities below about 5 X 10(6) M-1, the product of the required number of pepMHC complexes per target cell ("epitope density") and TCR affinity for pepMHC complexes is constant; therefore, over this range TCR affinities can be predicted from epitope densities (or vice versa). At higher receptor affinities ("affinity ceiling") the epitope density required for half-maximal lysis reaches a lower limit of less than 10 complexes per target cell.