This is the second paper in a series presenting case studies in modern large-scale constrained optimization [9]. In this paper, we consider the shape of a hanging chain, which, in equilibrium, minimizes the potential energy of the chain. In addition to the tutorial aspects of this paper, we also emphasize the importance of certain modeling issues such as convex vs. nonconvex formulations of giv...

A. Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S )) and the tame degree of S (denoted t(S )) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we ...

Air gap impulse discharge characteristics of railway tunnel-catenary at high-altitude are great significance to the electrical insulation design. The existing altitude correction methods for non-uniform air gaps not applicable tunnel-catenary. To investigate tunnel-catenary, a typical structural model was built in an artificial climate chamber. Different altitudes were simulated by adjusting at...

Accurate semantic segmentation results of the overhead catenary system (OCS) are significant for OCS component extraction and geometric parameter detection. Actually, scenes complex, density point cloud data obtained through Light Detection Ranging (LiDAR) scanning is uneven due to character difference components. However, inconsistent points, it challenging complete better with existing deep l...

Abstract As a critical component of ensuring the safe and stable operation trains, detection bird’s nests on rail catenary has always been essential. Low-resolution images lack labelled data, however, make it difficult to detect smaller (those occupying small pixels in input image). Previous solution relies manual online patrol or offline video playback, which severely limits efficiency. Previo...

We prove that the generic quantized coordinate ringOq(G) is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group G. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a...

The trajectory of a rider slowly traversing a zipline of fixed length and endpoints is computed, as a one-dimensional illustration of the curving of spacetime by a mass. As the rider’s weight increases from 0 to ∞, the trajectories uniformly interpolate between a catenary and an ellipse. These results are obtained by balancing forces in the horizontal and vertical directions, so that the treatm...

Let us consider a 3-numerical semigroup S = 〈a, b,N 〉. Given m ∈ S, the triple (x, y, z) ∈ N3 is a factorization of m in S if xa+ yb+ zN = m. This work is focused on finding the full set of factorizations of any m ∈ S and as an application we compute the catenary degree of S. To this end, we relate a 2D tessellation to S and we use it as a main tool.