Set-Invariance-based Interpretations for the L1 Performance of Nonlinear Systems

This paper is concerend with tackling the L 1 performance analysis problem of continuous and piecewise continuous nonlinear systems with non-unique solutions by using the involved arguments of set-invariance principles. More precisely, this paper derives a sufficient condition for the L 1 performance of continuous nonlinear systems in terms of the invariant set. However, because this sufficient condition intrinsically involves analytical representations of solutions of the differential equations corresponding to the nonlinear systems, this paper also establishes another sufficient condition for the L 1 performance by introducing the so-called extended invariance domain, in which it is not required to directly solving the nonlinear differential equations. These arguments associated with the L 1 performance analysis is further extended to the case of piecewise continuous nonlinear systems, and we obtain parallel results based on the set-invariance principles used for the continuous nonolinear systems. Finally, numerical examples are provided to demonstrate the effectiveness as well as the applicability of the overall results derived in this paper.

Image classification and cognition using contour curvature statistics

Although the higher order mechanisms behind object representation and classification in the visual system are still not well understood, there are hints that simple shape primitives such as “curviness” might activate neural activation and guide this process. Drawing on elementary invariance principles, we propose that a statistical geometric object, the probability distribution of the normalized contour curvatures (NCC) in the intensity field of a planar image, has the potential to represent and classify categories of objects. We show that NCC is sufficient for discriminating between cognitive categories such as animacy, size and type, and demonstrate the robustness of this metric to variation in illumination and viewpoint, consistent with neurobiological constraints and psychological experiments. A generative model for producing artificial images with the observed NCC distributions highlights the key features that our metric captures and just as importantly, those that it does not. More broadly, our study points to the need for statistical geometric approaches to cognition that build in both the statistics and the natural invariances of the sensory world.

Invariance principles in the theory and computation of transport coefficients

In this work, we elaborate on two recently discovered invariance principles, according to which transport coefficients are, to a large extent, independent of the microscopic definition of the densities and currents of the conserved quantities being transported (energy, momentum, mass, charge). The first such principle, gauge invariance, allows one to define a quantum adiabatic energy current from density-functional theory, from which the heat conductivity can be uniquely defined and computed using equilibrium ab initio molecular dynamics. When combined with a novel topological definition of atomic oxidation states, gauge invariance also sheds new light onto the mechanisms of charge transport in ionic conductors. The second principle, convective invariance, allows one to extend the analysis to multi-component systems. These invariance principles can be combined with new spectral analysis methods for the current time series to be fed into the Green–Kubo formula to obtain accurate estimates of transport coefficients from relatively short molecular dynamics simulations. Graphic abstract

On the Role of Self-similarity in the Dynamics of Effective Field Theory

The goal of this work is to show that the underlying symmetries of effective field theory can be traced to the onset of self-similarity. In particular, we argue that the scale-free structure of fractal geometry lies at the heart of invariance principles in classical and Quantum Field Theory.