A BPS Interpretation of Shape Invariance

We show that shape invariance appears when a quantum mechanical model is invariant under a centrally extended superalgebra endowed with an additional symmetry generator, which we dub the shift operator. The familiar mathematical and physical results of shape invariance then arise from the BPS structure associated with this shift operator. The shift operator also ensures that there is a one-to-one correspondence between the energy levels of such a model and the energies of the BPS-saturating states. These findings thus provide a more comprehensive algebraic setting for understanding shape invariance.Comment: 15 pages, 2 figures, LaTe

Free Energy of an Inhomogeneous Superconductor: a Wave Function Approach

A new method for calculating the free energy of an inhomogeneous superconductor is presented. This method is based on the quasiclassical limit (or Andreev approximation) of the Bogoliubov-de Gennes (or wave function) formulation of the theory of weakly coupled superconductors. The method is applicable to any pure bulk superconductor described by a pair potential with arbitrary spatial dependence, in the presence of supercurrents and external magnetic field. We find that both the local density of states and the free energy density of an inhomogeneous superconductor can be expressed in terms of the diagonal resolvent of the corresponding Andreev Hamiltonian, resolvent which obeys the so-called Gelfand-Dikii equation. Also, the connection between the well known Eilenberger equation for the quasiclassical Green's function and the less known Gelfand-Dikii equation for the diagonal resolvent of the Andreev Hamiltonian is established. These results are used to construct a general algorithm for calculating the (gauge invariant) gradient expansion of the free energy density of an inhomogeneous superconductor at arbitrary temperatures.Comment: REVTeX, 28 page

Research and Technology Challenges for Human Data Analysts in Future Safety Management Systems

Enabling new and novel concepts of operations for Advanced Air Mobility poses an important need to evolve current safety management systems (SMS) and is posited to be realized through advances in Machine Learning (ML) Data Sciences and Artificial Intelligence. The “In-time Aviation Safety Management System” (IASMS) concept of operations supports the need to evolve today’s SMS to become more tailorable, scalable, and interoperable in response to forecasted changes expected for the future airspace system. Key to IASMS is integration of proactive and predictive ML algorithms trained to provide “in time” detection and mitigation of hazards and emergent risks through new methods and novel data types. IASMS research and technology development includes human factors design considerations for these systems to include human-system teaming, innovations in human interfaces and management of complex digital data information, human-system interaction/model-based system engineering, and verification and validation for data assurance and trust