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.

Initiation process of earthquakes and its implications for seismic hazard reduction strategy.

For the average citizen and the public, "earthquake prediction" means "short-term prediction," a prediction of a specific earthquake on a relatively short time scale. Such prediction must specify the time, place, and magnitude of the earthquake in question with sufficiently high reliability. For this type of prediction, one must rely on some short-term precursors. Examinations of strain changes just before large earthquakes suggest that consistent detection of such precursory strain changes cannot be expected. Other precursory phenomena such as foreshocks and nonseismological anomalies do not occur consistently either. Thus, reliable short-term prediction would be very difficult. Although short-term predictions with large uncertainties could be useful for some areas if their social and economic environments can tolerate false alarms, such predictions would be impractical for most modern industrialized cities. A strategy for effective seismic hazard reduction is to take full advantage of the recent technical advancements in seismology, computers, and communication. In highly industrialized communities, rapid earthquake information is critically important for emergency services agencies, utilities, communications, financial companies, and media to make quick reports and damage estimates and to determine where emergency response is most needed. Long-term forecast, or prognosis, of earthquakes is important for development of realistic building codes, retrofitting existing structures, and land-use planning, but the distinction between short-term and long-term predictions needs to be clearly communicated to the public to avoid misunderstanding.

Intermediate-term earthquake prediction.

An earthquake of magnitude M and linear source dimension L(M) is preceded within a few years by certain patterns of seismicity in the magnitude range down to about (M - 3) in an area of linear dimension about 5L-10L. Prediction algorithms based on such patterns may allow one to predict approximately 80% of strong earthquakes with alarms occupying altogether 20-30% of the time-space considered. An area of alarm can be narrowed down to 2L-3L when observations include lower magnitudes, down to about (M - 4). In spite of their limited accuracy, such predictions open a possibility to prevent considerable damage. The following findings may provide for further development of prediction methods: (i) long-range correlations in fault system dynamics and accordingly large size of the areas over which different observed fields could be averaged and analyzed jointly, (ii) specific symptoms of an approaching strong earthquake, (iii) the partial similarity of these symptoms worldwide, (iv) the fact that some of them are not Earth specific: we probably encountered in seismicity the symptoms of instability common for a wide class of nonlinear systems.