On the existence of quantum representations for two dichotomic measurements

Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories.Comment: 16+7 pages, presentation improved and minor errors correcte

Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case - A simple proof for finite extension

We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding growth condition on the equation of state p=P(\rho) at \rho=0. These purely local conditions are slight generalizations to known such conditions.Comment: 20 page

Phenotypic switching of populations of cells in a stochastic environment

In biology phenotypic switching is a common bet-hedging strategy in the face of uncertain environmental conditions. Existing mathematical models often focus on periodically changing environments to determine the optimal phenotypic response. We focus on the case in which the environment switches randomly between discrete states. Starting from an individual-based model we derive stochastic differential equations to describe the dynamics, and obtain analytical expressions for the mean instantaneous growth rates based on the theory of piecewise deterministic Markov processes. We show that optimal phenotypic responses are non-trivial for slow and intermediate environmental processes, and systematically compare the cases of periodic and random environments. The best response to random switching is more likely to be heterogeneity than in the case of deterministic periodic environments, net growth rates tend to be higher under stochastic environmental dynamics. The combined system of environment and population of cells can be interpreted as host-pathogen interaction, in which the host tries to choose environmental switching so as to minimise growth of the pathogen, and in which the pathogen employs a phenotypic switching optimised to increase its growth rate. We discuss the existence of Nash-like mutual best-response scenarios for such host-pathogen games.Comment: 17 pages, 6 figure

Breaking Kelvin: Circulation conservation and vortex breakup in MHD at low Magnetic Prandtl Number

In this paper we examine the role of weak magnetic fields in breaking Kelvin's circulation theorem and in vortex breakup in two-dimensional magnetohydrodynamics for the physically important case of a low magnetic Prandtl number (low $Pm$) fluid. We consider three canonical inviscid solutions for the purely hydrodynamical problem, namely a Gaussian vortex, a circular vortex patch and an elliptical vortex patch. We examine how magnetic fields lead to an initial loss of circulation $\Gamma$ and attempt to derive scaling laws for the loss of circulation as a function of field strength and diffusion as measured by two non-dimensional parameters. We show that for all cases the loss of circulation depends on the integrated effects of the Lorentz force, with the patch cases leading to significantly greater circulation loss. For the case of the elliptical vortex the loss of circulation depends on the total area swept out by the rotating vortex and so this leads to more efficient circulation loss than for a circular vortex.Comment: 21 pages, 12 figure

A pairwise maximum entropy model describes energy landscape for spiral wave dynamics of cardiac fibrillation

Heart is an electrically-connected network. Spiral wave dynamics of cardiac fibrillation shows chaotic and disintegrated patterns while sinus rhythm shows synchronized excitation patterns. To determine functional interactions between cardiomyocytes during complex fibrillation states, we applied a pairwise maximum entropy model (MEM) to the sequential electrical activity maps acquired from the 2D computational simulation of human atrial fibrillation. Then, we constructed energy landscape and estimated hierarchical structure among the different local minima (attractors) to explain the dynamic properties of cardiac fibrillation. Four types of the wave dynamics were considered: sinus rhythm; single stable rotor; single rotor with wavebreak; and multiple wavelet. The MEM could describe all types of wave dynamics (both accuracy and reliability>0.9) except the multiple random wavelet. Both of the sinus rhythm and the single stable rotor showed relatively high pairwise interaction coefficients among the cardiomyocytes. Also, the local energy minima had relatively large basins and high energy barrier, showing stable attractor properties. However, in the single rotor with wavebreak, there were relatively low pairwise interaction coefficients and a similar number of the local minima separated by a relatively low energy barrier compared with the single stable rotor case. The energy landscape of the multiple wavelet consisted of a large number of the local minima separated by a relatively low energy barrier, showing unstable dynamics. These results indicate that the MEM provides information about local and global coherence among the cardiomyocytes beyond the simple structural connectivity. Energy landscape analysis can explain stability and transitional properties of complex dynamics of cardiac fibrillation, which might be determined by the presence of 'driver' such as sinus node or rotor.Comment: Presented at the 62nd Biophysical Society Annual Meeting, San Francisco, California, 201