Doubly-Special Relativity: Facts, Myths and Some Key Open Issues

I report, emphasizing some key open issues and some aspects that are particularly relevant for phenomenology, on the status of the development of "doubly-special" relativistic ("DSR") theories with both an observer-independent high-velocity scale and an observer-independent small-length/large-momentum scale, possibly relevant for the Planck-scale/quantum-gravity realm. I also give a true/false characterization of the structure of these theories. In particular, I discuss a DSR scenario without modification of the energy-momentum dispersion relation and without the $\kappa$-Poincar\'e Hopf algebra, a scenario with deformed Poincar\'e symmetries which is not a DSR scenario, some scenarios with both an invariant length scale and an invariant velocity scale which are not DSR scenarios, and a DSR scenario in which it is easy to verify that some observable relativistic (but non-special-relativistic) features are insensitive to possible nonlinear redefinitions of symmetry generators.Comment: This is the preprint version of a paper prepared for a special issue "Feature Papers: Symmetry Concepts and Applications" of the journal Symmetr

Nonintegrability, Chaos, and Complexity

Two-dimensional driven dissipative flows are generally integrable via a conservation law that is singular at equilibria. Nonintegrable dynamical systems are confined to n*3 dimensions. Even driven-dissipative deterministic dynamical systems that are critical, chaotic or complex have n-1 local time-independent conservation laws that can be used to simplify the geometric picture of the flow over as many consecutive time intervals as one likes. Those conserevation laws generally have either branch cuts, phase singularities, or both. The consequence of the existence of singular conservation laws for experimental data analysis, and also for the search for scale-invariant critical states via uncontrolled approximations in deterministic dynamical systems, is discussed. Finally, the expectation of ubiquity of scaling laws and universality classes in dynamics is contrasted with the possibility that the most interesting dynamics in nature may be nonscaling, nonuniversal, and to some degree computationally complex

Equal-area method for scalar conservation laws

We study one-dimensional conservation law. We develop a simple numerical method for computing the unique entropy admissible weak solution to the initial problem. The method basis on the equal-area principle and gives the solution for given time directly.Comment: 10 pages, 7 figure

Variational Principle underlying Scale Invariant Social Systems

MaxEnt's variational principle, in conjunction with Shannon's logarithmic information measure, yields only exponential functional forms in straightforward fashion. In this communication we show how to overcome this limitation via the incorporation, into the variational process, of suitable dynamical information. As a consequence, we are able to formulate a somewhat generalized Shannonian Maximum Entropy approach which provides a unifying "thermodynamic-like" explanation for the scale-invariant phenomena observed in social contexts, as city-population distributions. We confirm the MaxEnt predictions by means of numerical experiments with random walkers, and compare them with some empirical data