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.

Differential Regulation of Sugar-Sensitive Sucrose Synthases by Hypoxia and Anoxia Indicate Complementary Transcriptional and Posttranscriptional Responses1

The goal of this research was to resolve the hypoxic and anoxic responses of maize (Zea mays) sucrose (Suc) synthases known to differ in their sugar regulation. The two maize Suc synthase genes, Sus1 and Sh1, both respond to sugar and O2, and recent work suggests commonalities between these signaling systems. Maize seedlings (NK508 hybrid, W22 inbred, and an isogenic sh1-null mutant) were exposed to anoxic, hypoxic, and aerobic conditions (0, 3, and 21% O2, respectively), when primary roots had reached approximately 5 cm. One-centimeter tips were excised for analysis during the 48-h treatments. At the mRNA level, Sus1 was rapidly up-regulated by hypoxia (approximately 5-fold in 6 h), whereas anoxia had less effect. In contrast, Sh1 mRNA abundance increased strongly under anoxia (approximately 5-fold in 24 h) and was much less affected by hypoxia. At the enzyme level, total Suc synthase activity rose rapidly under hypoxia but showed little significant change during anoxia. The contributions of SUS1 and SH1 activities to these responses were dissected over time by comparing the sh1-null mutant with the isogenic wild type (Sus+, Sh1+). Sh1-dependent activity contributed most markedly to a rapid protein-level response consistently observed in the first 3 h, and, subsequently, to a long-term change mediated at the level of mRNA accumulation at 48 h. A complementary midterm rise in SUS1 activity varied in duration with genetic background. These data highlight the involvement of distinctly different genes and probable signal mechanisms under hypoxia and anoxia, and together with earlier work, show parallel induction of “feast and famine” Suc synthase genes by hypoxia and anoxia, respectively. In addition, complementary modes of transcriptional and posttranscriptional regulation are implicated by these data, and provide a mechanism for sequential contributions from the Sus1 and Sh1 genes during progressive onset of naturally occurring low-O2 events.