Modified SIR-model applied to covid-19, similarity solutions and projections to further development

The SIR-model is adapted to the covid-19 pandemic through a modification that consists in making the basic reproduction number variable. Independent of it, another reproduction number is introduced, which is defined similarly to the usual net reproduction number. Due to its simple analytic form, it enables a clear interpretation for all values. A further parameter, provisionally called acceleration parameter, is introduced and applied, which enables a more differentiated characterization of the infection number dynamics. By a variable transformation the 3 equations of the modified SIR-model can be reduced to 2. The latter are solved up to ordinary integrations. The solutions are evaluated for current situations, yielding a pretty good match with the data reported. Encouraged by this, a variety of possible future developments is examined, including linear and exponential growth of the infection numbers as well as sub- and super-exponential growth. In particular, the behavior of the two reproduction numbers and the acceleration parameter is studied, which in some cases leads to surprising results. With regard to the number of unreported infections it is shown, that from the solution for a special one solutions for others can be derived by similarity transformations.

Multiscale Electrolyte Transport Simulations for Lithium Ion Batteries

Establishing a link between atomistic processes and battery cell behavior is a major challenge for lithium ion batteries. Focusing on liquid electrolytes, we describe parameter-free molecular dynamics predictions of their mass and charge transport properties. The simulations agree quantitatively with experiments across the full range of relevant ion concentrations and for different electrolyte compositions. We introduce a simple analytic form to describe the transport properties. Our results are used in an extended Newman electrochemical model, including a cell temperature prediction. This multi-scale approach provides quantitative agreement between calculated and measured discharge voltage of a battery and enables the computational optimization of the electrolyte formulation.

Multiscale Electrolyte Transport Simulations for Lithium Ion Batteries

Establishing a link between atomistic processes and battery cell behavior is a major challenge for lithium ion batteries. Focusing on liquid electrolytes, we describe parameter-free molecular dynamics predictions of their mass and charge transport properties. The simulations agree quantitatively with experiments across the full range of relevant ion concentrations and for different electrolyte compositions. We introduce a simple analytic form to describe the transport properties. Our results are used in an extended Newman electrochemical model, including a cell temperature prediction. This multi-scale approach provides quantitative agreement between calculated and measured discharge voltage of a battery and enables the computational optimization of the electrolyte formulation.

Scale-invariant singularity of the surface quasigeostrophic patch

Numerical simulations of the surface quasigeostrophic patch indicate the development of a scale-invariant singularity of the boundary curvature in finite time, with some evidence of universality across a variety of initial conditions. At the time of singularity, boundary segments are shown to possess an exact and simple analytic form, described by branches of a logarithmic spiral centred on the point of singularity. The angles between the branches depend non-trivially on the shape of the smooth connecting boundary as the singularity is approached, but are independent of the global boundary.

Equation of state and thermal expansion of metals with FCC structure: Application to Cu, Al and Ni

The equation of state, the expressions of lattice parameter and thermal expansion coefficient in general form are obtained by the statistical moment method. Applying to Cu , Al and Ni metals, we determine these properties in simple analytic form for each metal. Numerical results for the thermal expansion coefficient of these metals in different temperatures and pressures are in good agreement with experiments.

LONG TERM EVOLUTION OF MAGNETIC TURBULENCE IN RELATIVISTIC COLLISIONLESS SHOCKS

We study the long term evolution of magnetic fields generated by an initially unmagnetized collisionless relativistic e+e- shock. Our 2D particle-in-cell numerical simulations show that downstream of such a Weibel-mediated shock, particle distributions are approximately isotropic, relativistic Maxwellians, and the magnetic turbulence is highly intermittent spatially, non-propagating, and decaying. Using linear kinetic theory, we find a simple analytic form for these damping rates. Our theory predicts that the overall magnetic energy decays as (ωp t)-q with q ~ 1, which compares favorably with simulations, but predicts overly rapid damping of short-wavelength modes. The magnetic trapping of particles within the magnetic structures may be the origin of this discrepancy. We conclude that initially unmagnetized relativistic shocks in electron-positron plasmas are unable to form persistent downstream magnetic fields. These results put interesting constraints on synchrotron models for the prompt and afterglow emission from GRBs.

Why Is Climate Sensitivity So Unpredictable?

Uncertainties in projections of future climate change have not lessened substantially in past decades. Both models and observations yield broad probability distributions for long-term increases in global mean temperature expected from the doubling of atmospheric carbon dioxide, with small but finite probabilities of very large increases. We show that the shape of these probability distributions is an inevitable and general consequence of the nature of the climate system, and we derive a simple analytic form for the shape that fits recent published distributions very well. We show that the breadth of the distribution and, in particular, the probability of large temperature increases are relatively insensitive to decreases in uncertainties associated with the underlying climate processes.

Error estimates for the fast multipole method. I. The two-dimensional case

The Greengard-Rokhlin algorithm is a new and interesting method for computing long-range interactions in particle systems. Although the method already has been implemented and claimed to be superior to traditional and other methods, no reliable estimates of the size of the error of the method have been given. We illustrate what the error actually is for the two-dimensional case, and derive an estimate for it. The estimate has a simple analytic form which will allow its use in tuning the algorithm for best efficiency.