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.

On the controllability of distributed systems

To “control” a system is to make it behave (hopefully) according to our “wishes,” in a way compatible with safety and ethics, at the least possible cost. The systems considered here are distributed—i.e., governed (modeled) by partial differential equations (PDEs) of evolution. Our “wish” is to drive the system in a given time, by an adequate choice of the controls, from a given initial state to a final given state, which is the target. If this can be achieved (respectively, if we can reach any “neighborhood” of the target) the system, with the controls at our disposal, is exactly (respectively, approximately) controllable. A very general (and fuzzy) idea is that the more a system is “unstable” (chaotic, turbulent) the “simplest,” or the “cheapest,” it is to achieve exact or approximate controllability. When the PDEs are the Navier–Stokes equations, it leads to conjectures, which are presented and explained. Recent results, reported in this expository paper, essentially prove the conjectures in two space dimensions. In three space dimensions, a large number of new questions arise, some new results support (without proving) the conjectures, such as generic controllability and cases of decrease of cost of control when the instability increases. Short comments are made on models arising in climatology, thermoelasticity, non-Newtonian fluids, and molecular chemistry. The Introduction of the paper and the first part of all sections are not technical. Many open questions are mentioned in the text.

Selenophosphate synthetase: detection in extracts of rat tissues by immunoblot assay and partial purification of the enzyme from the archaean Methanococcus vannielii.

In Escherichia coli and Salmonella typhimurium it has been shown that selenophosphate serves as the selenium donor for the conversion of seryl-tRNA to selenocysteyl-tRNA and for the synthesis of 2-selenouridine, a modified nucleoside present in tRNAs. Although selenocysteyl-tRNA also is formed in eukaryotes and is used for the specific insertion of selenocysteine into proteins, the precise mechanism of its biosynthesis from seryl-tRNA in these systems is not known. Because selenophosphate is extremely oxygen labile and difficult to identify in biological systems, we used an immunological approach to detect the possible presence of selenophosphate synthetase in mammalian tissues. With antibodies elicited to E. coli selenophosphate synthetase the enzyme was detected in extracts of rat brain, liver, kidney, and lung by immunoblotting. Especially high levels were detected in Methanococcus vannielii, a member of the domain Archaea, and the enzyme was partially purified from this source. It seems likely that the use of selenophosphate as a selenium donor is widespread in biological systems.

Sticky ions in biological systems.

Aqueous gel sieving chromatography on Sephadex G-10 of the Group IA cations (Li+, Na+, K+, Rb+, Cs+) plus NH4+ as the Cl- salts, in combination with previous results for the halide anions (F-, Cl-, Br-, I-) as the Na+ salts [Washabaugh, M.W. & Collins, K.D. (1986) J. Biol. Chem. 261, 12477-12485], leads to the following conclusions. (i) The small monovalent ions (Li+, Na+, F-) flow through the gel with water molecules attached, whereas the large monovalent ions (K+, Rb+, Cs+, Cl-, Br-, I-) adsorb to the nonpolar surface of the gel, a process requiring partial dehydration of the ion and implying that these ions bind the immediately adjacent water molecules weakly. (ii) The transition from strong to weak hydration occurs at a radius of about 1.78 A for the monovalent anions, compared with a radius of about 1.06 A for the monovalent cations (using ionic radii), indicating that the anions are more strongly hydrated than the cations for a given charge density. (iii) The anions show larger deviations from ideal behavior (an elution position corresponding to the anhydrous molecular weight) than do the cations and dominate the chromatographic behavior of the neutral salts. These results are interpreted to mean that weakly hydrated ions (chaotropes) are "pushed" onto weakly hydrated surfaces by strong water-water interactions and that the transition from strong ionic hydration to weak ionic hydration occurs where the strength of ion-water interactions approximately equals the strength of water-water interactions in bulk solution.

Rescue, propagation, and partial purification of a helper virus-dependent adenovirus vector.

Adenoviral vectors are widely used as highly efficient gene transfer vehicles in a variety of biological research strategies including human gene therapy. One of the limitations of the currently available adenoviral vector system is the presence of the majority of the viral genome in the vector, resulting in leaky expression of viral genes particularly at high multiplicity of infection and limited cloning capacity of exogenous sequences. As a first step to overcome this problem, we attempted to rescue a defective human adenovirus serotype 5 DNA, which had an essential region of the viral genome (L1, L2, VAI + II, pTP) deleted and replaced with an indicator gene. In the presence of wild-type adenovirus as a helper, this DNA was packaged and propagated as transducing viral particles. After several rounds of amplification, the titer of the recombinant virus reached at least 4 x 10(6) transducing particles per ml. The recombinant virus could be partially purified from the helper virus by CsCl equilibrium density-gradient centrifugation. The structure of the recombinant virus around the marker gene remained intact after serial propagation, while the pBR sequence inserted in the E1 region was deleted from the recombinant virus. Our results suggest that it should be possible to develop a helper-dependent adenoviral vector, which does not encode any viral proteins, as an alternative to the currently available adenoviral vector systems.