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.

Molecular origin of the mosaic sequence arrangements of higher primate α-globin duplication units

The human adult α-globin locus consists of three pairs of homology blocks (X, Y, and Z) interspersed with three nonhomology blocks (I, II, and III), and three Alu family repeats, Alu1, Alu2, and Alu3. It has been suggested that an ancient primate α-globin-containing unit was ancestral to the X, Y, and Z and the Alu1/Alu2 repeats. However, the evolutionary origin of the three nonhomologous blocks has remained obscure. We have now analyzed the sequence organization of the entire adult α-globin locus of gibbon (Hylobates lar). DNA segments homologous to human block I occur in both duplication units of the gibbon α-globin locus. Detailed interspecies sequence comparisons suggest that nonhomologous blocks I and II, as well as another sequence, IV, were all part of the ancestral α-globin-containing unit prior to its tandem duplication. However, sometime thereafter, block I was deleted from the human α1-globin-containing unit, and block II was also deleted from the α2-globin-containing unit in both human and gibbon. These were probably independent events both mediated by independent illegitimate recombination processes. Interestingly, the end points of these deletions coincide with potential insertion sites of Alu family repeats. These results suggest that the shaping of DNA segments in eukaryotic genomes involved the retroposition of repetitive DNA elements in conjunction with simple DNA recombination processes.

Hyperplane arrangements, interval orders, and trees.

A hyperplane arrangement is a finite set of hyperplanes in a real affine space. An especially important arrangement is the braid arrangement, which is the set of all hyperplanes xi - xj = 1, 1