Although General Relativity is the classic example of a physical theory based on differential geometry, the momentum tensor is the only part of the field equation that is not derived from or interpreted with differential geometry. This work extends General Relativity and Einstein-Cartan theory by augmenting the Poincaré group with projective (special) conformal transformations, which are translations at conformal infinity. Momentum becomes a part of the differential geometry of spacetime. The Lie algebra of these transformations is represented by vectorfields on an associated Minkowski fiber space. Variation of projective conformal scalar curvature generates a 2-index tensor that serves as linear momentum in the field equations of General Relativity. The computation yields a constructive realization of Mach’s principle: local inertia is determined by local motion relative to mass at conformal infinity in each fiber. The vectorfields have a cellular structure that is similar to that of turbulent fluids.
This paper discusses the feasibility of thin-shell wormholes in spacetimes of embedding class one admitting a one-parameter group of conformal motions. It is shown that the surface energy density σis positive, while the surface pressure is negative, resulting in , thereby signaling a violation of the null energy condition, a necessary condition for holding a wormhole open. For a Morris-Thorne wormhole, matter that violates the null energy condition is referred to as “exotic”. For the thin-shell wormholes in this paper, however, the violation has a physical explanation since it is a direct consequence of the embedding theory in conjunction with the assumption of conformal symmetry. These properties avoid the need to hypothesize the existence of the highly problematical exotic matter.
The scientific community recognizes the seriousness of rockbursts and the need for effective mitigation measures.The literature reports various successful applications of machine learning(ML)models for rockburst assessment;however,a significant question remains unanswered:How reliable are these models,and at what confidence level are classifications made?Typically,ML models output single rockburst grade even in the face of intricate and out-of-distribution samples,without any associated confidence value.Given the susceptibility of ML models to errors,it becomes imperative to quantify their uncertainty to prevent consequential failures.To address this issue,we propose a conformal prediction(CP)framework built on traditional ML models(extreme gradient boosting and random forest)to generate valid classifications of rockburst while producing a measure of confidence for its output.The proposed framework guarantees marginal coverage and,in most cases,conditional coverage on the test dataset.The CP was evaluated on a rockburst case in the Sanshandao Gold Mine in China,where it achieved high coverage and efficiency at applicable confidence levels.Significantly,the CP identified several“confident”classifications from the traditional ML model as unreliable,necessitating expert verification for informed decision-making.The proposed framework improves the reliability and accuracy of rockburst assessments,with the potential to bolster user confidence.
In this paper,an effective algorithm for optimizing the subarray of conformal arrays is proposed.The method first divides theconformal array into several first-level subarrays.It uses the X algorithm to solve the feasible solution of first-level subarray tiling and employs the particle swarm algorithm to optimize the conformal array subarray tiling scheme with the maximum entropy of the planar mapping as the fitness function.Subsequently,convex optimization is applied to optimize the subarray amplitude phase.Data results verify that the method can effectively find the optimal conformal array tiling scheme.
