Integrable and Superintegrable Systems with Higher Order Integrals of Motion: Master Function Formalism

We construct two-dimensional integrable and superintegrable systems in terms of the master function formalism and relate them to Mielnik’s and Marquette’s construction in supersymmetric quantum mechanics. For two different cases of the master functions, we obtain two different two-dimensional superintegrable systems with higher order integrals of motion.

Master Function for Analytical Determination of the Interlayer Bond Shear Stiffness and Fatigue Functions for Asphalt Pavements

In order to determine the shear stiffness at the interface between asphalt layers and to take into account the interactions of repeated traffic loading, acceleration and braking processes as well as temperature influence, a complicated apparatus for cycling testing of the interlayer bond (CTIB) has been developed. An extensive experimental procedure has been created to include all factors that influence the interlayer bond. Using the experimental results, a master function for the analytical assessment of the shear stiffness has been established. The regression which approximates most accurately the experimentally determined shear stiffness values is the sigmoid function. Through implementation of the master function into a finite element program the fatigue status of asphalt pavements, which is affected by the interlayer bond of different quality, have been calculated over the service life of 30 years using the German method for computational design according to RDO Asphalt 09. The results presented below are based on the results of IGF project “Cyclic Shear Stiffness and Shear Fatigue Testing for Evaluation and Optimization of Interlayer Bond in Asphalt Pavements”, supported by the Association of Industrial Research Communities (AIF) of the German Asphalt Institute (DAI) in cooperation with TU Braunschweig.

Can A Robot Have Free Will?

Using insights from cybernetics and an information-based understanding of biological systems, a precise, scientifically inspired, definition of free-will is offered and the essential requirements for an agent to possess it in principle are set out. These are: a) there must be a self to self-determine; b) there must be a non-zero probability of more than one option being enacted; c) there must be an internal means of choosing among options (which is not merely random, since randomness is not a choice). For (a) to be fulfilled, the agent of self-determination must be organisationally closed (a `Kantian whole'). For (c) to be fulfilled: d) options must be generated from an internal model of the self which can calculate future states contingent on possible responses; e) choosing among these options requires their evaluation using an internally generated goal defined on an objective function representing the overall `master function' of the agent and f) for `deep free-will', at least two nested levels of choice and goal (d-e) must be enacted by the agent. The agent must also be able to enact its choice in physical reality. The only systems known to meet all these criteria are living organisms, not just humans, but a wide range of organisms. The main impediment to free-will in present-day artificial robots, is their lack of being a Kantian whole. Consciousness does not seem to be a requirement and the minimum complexity for a free-will system may be quite low and include relatively simple life-forms that are at least able to learn.

Exact Solution of the Curved Dirac Equation in Polar Coordinates: Master Function Approach

We show that the (2+1) curved Dirac equation in polar coordinates can be transformed into Schrodinger-like differential equation for upper spinor component. We compare this equation with the Schrodinger equation derived from shape invariance property of second order differential equations of mathematical physics. This formalism enables us to determine the electrostatic potential and relativistic energy in terms of master function and corresponding weight function. We also obtain the spinor wave function in terms of orthogonal polynomials.

Critical Points and Resonance of Hyperplane Arrangements

If is a master function corresponding to a hyperplane arrangement 𝒜 and a collection of weights ⋋, we investigate the relationship between the critical set of , the variety defined by the vanishing of the one-form ⩊⋋ = d log , and the resonance of ⋋. For arrangements satisfying certain conditions, we show that if ⋋ is resonant in dimension p, then the critical set of has codimension at most p. These include all free arrangements and all rank 3 arrangements.