Fast and numerically stable circle fit

We develop a new algorithm for fitting circles that does not have drawbacks commonly found in existing circle fits. Our fit achieves ultimate accuracy (to machine precision), avoids divergence, and is numerically stable even when fitting circles get arbitrary large. Lastly, our algorithm takes less than 10 iterations to converge, on average.Comment: 16 page

Electrical current in Sinai billiards under general small forces

The Lorentz gas of $\mathbb{Z}^2$-periodic scatterers (or the so called Sinai billiards) can be used to model motion of electrons on an ionized medal. We investigate the linear response for the system under various external forces (during both the flight and the collision). We give some characterizations under which the forced system is time-reversible, and derive an estimate of the electrical current generated by the forced system. Moreover, applying Pesin entropy formula and Young dimension formula, we get several characterizations of the non-equilibrium steady state of the forced system.Comment: 19 page

Log-periodic drift oscillations in self-similar billiards

We study a particle moving at unit speed in a self-similar Lorentz billiard channel; the latter consists of an infinite sequence of cells which are identical in shape but growing exponentially in size, from left to right. We present numerical computation of the drift term in this system and establish the logarithmic periodicity of the corrections to the average drift

Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas

We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions.Similarities and differences are noted.Comment: 24 pages with 9 figure

Cytokine Profile as a Marker of Cell Damage and Immune Dysfunction after Spinal Cord Injury

The study reviews findings of the recent experiments designed to investigate cytokine profile after a spinal cord injury. The role of key cytokines was assessed in the formation of cellular response to trauma. The specific immunopathogenic interaction of the nervous and immune systems in the immediate and chronic post-traumatic periods is summarized. The practicality of a step-by-step approach to assessing the cytokine profile in spinal cord injury is shown, the need to take into account the combination of pathogenetic and protective components in the implementation regulatory effects of individual cytokines, their integration into regenerative processes in the damaged spinal cord, which allows a rational approach to the organization of the treatment process and the development of new medicines