Abstract:
The deterministic description of a wave of solution particle of efavirenz is given. Simulated pharmacokinetic data points from patients on efavirenz are used. The one dimensional wave equation is used to infer on transfer of vibrations due to tension between solution particles. The work investigates movement using wave analogy, but in a different variable space. Two important movement fluxes of a wave are derived an attracting one identified as tension conductivity and a dispersing one identified as tension diffusivity. The Wave Equation can be used to describe another spin-off movement flux formed induced by vibrations in solution particle.

Abstract:
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional fractional differential equations. A first application is discussed in relation to the d-dimensional fractional advection-dispersion equation. We also study the connection with multidimensional L\'evy processes.

Abstract:
A fractional diffusion equation with advection term is rigorously derived from a kinetic transport model with a linear turning operator, featuring a fat-tailed equilibrium distribution and a small directional bias due to a given vector field. The analysis is based on bounds derived by relative entropy inequalities and on two recently developed approaches for the macroscopic limit: a Fourier-Laplace transform method for spatially homogeneous data and the so called moment method, based on a modified test function.

Abstract:
We study a quasi-two-dimensional electrostatic drift kinetic system as a model for near-marginal ion temperature gradient (ITG) driven turbulence. A proof is given of the nonlinear stability of this system under conditions of linear stability. This proof is achieved using a transformation that diagonalizes the linear dynamics and also commutes with nonlinear $E\times B$ advection. For the case when linear instability is present, a corollary is found that forbids nonlinear energy transfer between appropriately defined sets of stable and unstable modes. It is speculated that this may explain the preservation of linear eigenmodes in nonlinear gyrokinetic simulations. Based on this property, a dimensionally reduced ($\infty\times\infty \rightarrow 1$) system is derived that may be useful for understanding dynamics around the critical gradient of Dimits.

Abstract:
A novel asymptotic matching procedure is developed to revisit the problem of magnetic island saturation in case of a finite current gradient at the rational surface. Nonlinear dispersion relation is derived for saturated magnetic island. It is shown that arbitrary normalization factors that were present in previous theories are fully specified with the asymptotic matching.

Abstract:
Within the traditional frame of reduced MHD, a new rigorous perturbation expansion provides the equation ruling the nonlinear growth and saturation of the tearing mode for any current gradient. The small parameter is the magnetic island width w. For the first time, the final equation displays at once terms of order w ln(1/w) and w which have the same magnitude for practical purposes; two new O(w) terms involve the current gradient. The technique is applicable to the case of an external forcing. The solution for a static forcing is computed explicitly and it exhibits three physical regimes.

Abstract:
We examine numerically the storage capacity and the behaviour near saturation of an attractor neural network consisting of bistable elements with an adjustable coupling strength, the Bistable Gradient Network (BGN). For strong coupling, we find evidence of a first-order "memory blackout" phase transition as in the Hopfield network. For weak coupling, on the other hand, there is no evidence of such a transition and memorized patterns can be stable even at high levels of loading. The enhanced storage capacity comes, however, at the cost of imperfect retrieval of the patterns from corrupted versions.

Abstract:
A scaling theory of long-wavelength electrostatic turbulence in a magnetised, weakly collisional plasma (e.g., drift-wave turbulence driven by temperature gradients) is proposed, with account taken both of the nonlinear advection of the perturbed particle distribution by fluctuating ExB flows and of its phase mixing, which is caused by the streaming of the particles along the mean magnetic field and, in a linear problem, would lead to Landau damping. A consistent theory is constructed in which very little free energy leaks into high velocity moments of the distribution, rendering the turbulent cascade in the energetically relevant part of the wave-number space essentially fluid-like. The velocity-space spectra of free energy expressed in terms of Hermite-moment orders are steep power laws and so the free-energy content of the phase space does not diverge at infinitesimal collisionality (while it does for a linear problem); collisional heating due to long-wavelength perturbations vanishes in this limit (also in contrast with the linear problem, in which it occurs at the finite rate equal to the Landau-damping rate). The ability of the free energy to stay in the low velocity moments of the distribution is facilitated by the "anti-phase-mixing" effect, whose presence in the nonlinear system is due to the stochastic version of the plasma echo (the advecting velocity couples the phase-mixing and anti-phase-mixing perturbations). The partitioning of the wave-number space between the (energetically dominant) region where this is the case and the region where linear phase mixing wins is governed by the "critical balance" between linear and nonlinear timescales (which for high Hermite moments splits into two thresholds, one demarcating the wave-number region where phase mixing predominates, the other where plasma echo does).

Abstract:
Density functional theory (DFT) is notorious for the absence of gradient corrections to the two-dimensional (2D) Thomas-Fermi kinetic-energy functional; it is widely accepted that the 2D analog of the 3D von Weizs\"acker correction vanishes, together with all higher-order corrections. Contrary to this long-held belief, we show that the leading correction to the kinetic energy does not vanish, is unambiguous, and contributes perturbatively to the total energy. This insight emerges naturally in a simple extension of standard DFT, which has the effective potential energy as a functional variable on equal footing with the single-particle density.

Abstract:
There is a significant nonlinear correlation between the Eddington ratio (L_bol=L_Edd) and the Eddington-scaled kinetic power (L_kin=L_Edd) of jets in low luminosity active galactic nuclei (AGNs) (Merloni & Heinz). It is believed that these low luminosity AGNs contain advection dominated accretion flows (ADAFs). We adopt the ADAF model developed by Li & Cao, in which the global dynamics of ADAFs with magnetically driven outflows is derived numerically, to explore the relation between bolometric luminosity and kinetic power of jets. We find that the observed relation, L_kin/L_Edd ~ (L_bol=L_Edd)^0.49, can be well reproduced by the model calculations with reasonable parameters for ADAFs with magnetically driven outflows. Our model calculations is always consistent with the slope of the correlation independent of the values of the parameters adopted. Compared with the observations, our results show that over 60% of the accreted gas at the outer radius escapes from the accretion disc in a wind before the gas falls into the black holes. The observed correlation between Eddington-scaled kinetic power and Bondi power can also be qualitatively reproduced by our model calculations. Our results show that the mechanical efficiency varies from 10^-2 ~ 10^-3, which is roughly consistent with that required in AGN feedback simulations.