On the computational efficiency of subgradient methods: a case study with Lagrangian bounds

Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications. We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables. We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different LDs of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices. Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs.     

Derivation of the Dirac Equation by Conformal Differential Geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.     

Triptycene‐Roofed Quinoxaline Cavitands for the Supramolecular Detection of BTEX in Air

Two novel triptycene quinoxaline cavitands ( DiTriptyQxCav and MonoTriptyQxCav ) have been designed, synthesized, and applied in the supramolecular detection of benzene, toluene, ethylbenzene, and xylenes (BTEX) in air. The complexation properties of the two cavitands towards aromatics in the solid state are strengthened by the presence of the triptycene moieties at the upper rim of the tetraquinoxaline walls, promoting the confinement of the aromatic hydrocarbons within the cavity. The two cavitands were used as fiber coatings for solid‐phase microextraction (SPME) BTEX monitoring in air. The best performances in terms of enrichment factors, selectivity, and LOD (limit of detection) values were obtained by using the DiTriptyQxCav coating. The corresponding SPME fiber was successfully tested under real urban monitoring conditions, outperforming the commercial divinylbenzene–Carboxen–polydimethylsiloxane (DVB–CAR–PDMS) fiber in BTEX adsorption.     

Extensions of vector bundles on P1

We determine the triples F,G,H of vector bundles of finite rank on P¹ which may fit in an exact sequence 0→F→G→H→0.     

The Cauchy Problem for Hyperbolic Operators with Double Characteristics in Presence of Transition

We study the C ^∞ well-posedness of the Cauchy problem for a class of hyperbolic second order operators with double characteristics in presence of geometric transitions.