In this paper, we introduce a novel integration method of Kardar-Parisi-Zhang (KPZ) equation. It has always been known that if during the discrete integration of the KPZ equation the nearest-neighbor height-difference exceeds a critical value, an instability appears and the integration diverges. One way to avoid these instabilities is to replace the KPZ nonlinear-term by a function of the same term that depends on a single adjustable parameter which is able to control pillars or grooves growing on the interface. Here, we propose a different integration method which consists of directly limiting the value taken by the KPZ nonlinearity, thereby imposing a restriction rule that is applied in each integration time-step, as if it were the growth rule of a restricted discrete model, e.g. restricted-solid-on-solid (RSOS). Taking the discrete KPZ equation with restrictions to its dimensionless version, the integration depends on three parameters: the coupling constant g, the inverse of the time-step k, and the restriction constant ε which is chosen to eliminate divergences while keeping all the properties of the continuous KPZ equation. We study in detail the conditions in the parameters' space that avoids divergences in the 1-dimensional integration and reproduce the scaling properties of the continuous KPZ with a particular parameter set. We apply the tested methodology to the d-dimensional case (d = 3, 4) with the purpose of obtaining the growth exponent β, by establishing the conditions of the coupling constant g under which we recover known values reached by other authors, in particular for the RSOS model. This method allows us to infer that d = 4 is not the critical dimension of the KPZ universality class, where the strong-coupling phase dissapears.

Recently, the interest on theoretical and experimental studies of dynamic properties of the magnetic domain wall (MDW) of ferromagnetic thin films with disorder placed in an external magnetic field has increased. In order to study global and local measurable observables, we consider the (1 + 1)-dimensional model introduced by Buceta and Muraca [Physica A 390 (2011) 4192], based on rules of evolution that describe the MDW avalanches. From the values of the roughness exponents, global ζ, local ζ loc , and spectral ζ s , obtained from the global interface width, hight-difference correlation function and structure function, respectively, recent works have concluded that the universality classes should be analyzed in the context of the anomalous scaling theory. We show that the model is included in the group of systems with intrinsic anomalous scaling (ζ ≃ 1.5, ζ loc = ζ s ≃ 0.5), and that the surface of the MDW is multi-affine. With these results, we hope to establish in short term the scaling relations that verify the critical exponents of the model, including the dynamic exponent z, the exponents of the distributions of avalanche-size τ and -duration α, among others.

The COVID-19 pandemic has challenged authorities at different levels of government administration around the globe. When faced with diseases of this severity, it is useful for the authorities to have prediction tools to estimate in advance the impact on the health system as well as the human, material, and economic resources that will be necessary. In this paper, we construct an extended Susceptible-Exposed-Infected-Recovered model that incorporates the social structure of Mar del Plata, the 4°most inhabited city in Argentina and head of the Municipality of General Pueyrredón. Moreover, we consider detailed partitions of infected individuals according to the illness severity, as well as data of local health resources, to bring predictions closer to the local reality. Tuning the corresponding epidemic parameters for COVID-19, we study an alternating quarantine strategy: a part of the population can circulate without restrictions at any time, while the rest is equally divided into two groups and goes on successive periods of normal activity and lockdown, each one with a duration of days. We also implement a random testing strategy with a threshold over the population. We found that is a good choice for the quarantine strategy since it reduces the infected population and, conveniently, it suits a weekly schedule. Focusing on the health system, projecting from the situation as of September 30, we foresee a difficulty to avoid saturation of the available ICU, given the extremely low levels of mobility that would be required. In the worst case, our model estimates that four thousand deaths would occur, of which 30% could be avoided with proper medical attention. Nonetheless, we found that aggressive testing would allow an increase in the percentage of people that can circulate without restrictions, and the medical facilities to deal with the additional critical patients would be relatively low.