Formalization of Ring Theory in PVS

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings.

     

Automated classification of simulated wind field patterns from multiphysics ensemble forecasts

In this study, we have proposed an automated classification approach to identify meaningful patterns in wind field data. Utilizing an extensive simulated wind database, we have demonstrated that the proposed approach can identify low‐level jets, near‐uniform profiles, and other patterns in a reliable manner. We have studied the dependence of these wind profile patterns on locations (eg, offshore vs onshore), seasons, and diurnal cycles. Furthermore, we have found that the probability distributions of some of the patterns depend on the underlying planetary boundary layer schemes in a significant way. The future potential of the proposed approach in wind resource assessment and, more generally, in mesoscale model parameterization improvement is touched upon in this paper.