Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

In a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

Parametric integration of design features selection with recording of construction of working machines

Adaptation of the graphic program for constructing of a specific class of technical means, being the specialty of the design and construction office, is the basic challenge of the market economy. This office that prepares the offer and then the competitive construction of the technical means in the shortest possible time as a result obtains the order. This effect is enabled by graphic software applications.

Semi-empirical relationships to assess the seismic performance of slopes from an updated version of the Italian seismic database

Seismic performance of slopes can be assessed through displacement-based procedures where earthquake-induced displacements are usually computed following Newmark-type calculations. These can be adopted to perform a parametric integration of earthquake records to evaluate permanent displacements for different slope characteristics and seismic input properties. Several semi-empirical relationships can be obtained for different purposes: obtaining site-specific displacement hazard curves following a fully-probabilistic approach, to assess the seismic risk associated with the slope; providing semi-empirical models within a deterministic framework, where the seismic-induced permanent displacement is compared with threshold values related to different levels of seismic performance; calibrating the seismic coefficient to be used in pseudo-static calculations, where a safety factor against limit conditions is computed. In this paper, semi-empirical relationships are obtained as a result of a parametric integration of an updated version of the Italian strong-motion database, that, in turn, is described and compared to older versions of the database and to well-known ground motion prediction equations. Permanent displacement is expressed as a function of either ground motion parameters, for a given yield seismic coefficient of the slope, or of both ground motion parameters and the seismic coefficient. The first are meant to be used as a tool to develop site-specific displacement hazard curves, while the last can be used to evaluate earthquake-induced slope displacements, as well as to calibrate the seismic coefficient to be used in a pseudo-static analysis. Influence of the vertical component of seismic motion on these semi-empirical relationships is also assessed.