Positron nonextensivity effect on the propagation of dust ion acoustic Gardner waves

Propagation of dust ion-acoustic (DIA) Gardner wave in a dusty electron–positron–ion (e–p–i) plasma is investigated. This plasma consists of q-distributed electrons and positrons, warm ions, and dust grains. The effects of the electron nonextensivity, positron nonextensivity, and fractional parameter on the properties of DIA Gardner wave are investigated. Space fractional Gardner (SFG) equation is derived using the semi inverse technique. An efficient modified G′/G-expansion method is presented to solve the SFG equation. It is found that the amplitude of the DIA Gardner wave increases with an increase in space fractional parameter β $\left(\beta \right)$ and spatial parameter ζ $\left(\zeta \right)$ . On other hands, the DIA Gardner wave shape can be modulated using the space fractional parameter β $\left(\beta \right)$ . Our results may help understand the astrophysical environments such as star magnetospheres, solar flares, and galactic nuclei.

Novel MCMC methods for Bayesian inference of spatial parameter fields

<p>Geostatistical inference (or inversion) methods are commonly used to estimate the spatial distribution of heterogeneous soil properties (e.g., hydraulic conductivity) from indirect measurements (e.g., piezometric heads). One approach is to use Bayesian inversion to combine prior assumptions (prior models) with indirect measurements to predict soil parameters and their uncertainty, which can be expressed in form of a posterior parameter distribution. This approach is mathematically rigorous and elegant, but has a disadvantage. In realistic settings, analytical solutions do not exist, and numerical evaluation via Markov chain Monte Carlo (MCMC) methods can become computationally prohibitive. Especially when treating spatially distributed parameters for heterogeneous materials, constructing efficient MCMC methods is a major challenge.</p><p>Here, we present two novel MCMC methods that extend and combine existing MCMC algorithms to speed up convergence for spatial parameter fields. First, we present the<em> sequential pCN-MCMC</em>, which is a combination of the <em>sequential Gibbs sampler</em>, and the <em>pCN-MCMC</em>. This <em>sequential pCN-MCMC</em> is more efficient (faster convergence) than existing methods. It can be used for Bayesian inversion of multi-Gaussian prior models, often used in single-facies systems. Second, we present the <em>parallel-tempering sequential Gibbs MCMC</em>. This MCMC variant enables realistic inversion of multi-facies systems. By this, we mean systems with several facies in which we model the spatial position of facies (via training images and multiple point geostatistics) and the internal heterogeneity per facies (via multi-Gaussian fields). The proposed MCMC version is the first efficient method to find the posterior parameter distribution for such multi-facies systems with internal heterogeneities.</p><p>We demonstrate the applicability and efficiency of the two proposed methods on hydro-geological synthetic test problems and show that they outperform existing state of the art MCMC methods. With the two proposed MCMCs, we enable modellers to perform (1) faster Bayesian inversion of multi-Gaussian random fields for single-facies systems and (2) Bayesian inversion of more realistic fields for multi-facies systems with internal heterogeneity at affordable computational effort.</p>

Stochastic maximum principle for systems driven by local martingales with spatial parameters

<p style='text-indent:20px;'>We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.</p>

Масштабный эффект при автоволновой пластической деформации

The relationship between the spatial parameter of the development of localized plastic flow (the length of the autowave of localized plasticity) and the length of the deformed sample is studied. In experiments conducted on polycrystalline samples from an alloy of zirconium and technically pure aluminum, the logarithmic law of the relationship of these quantities, acting at the stage of Taylor parabolic work hardening, was established.

Martingale decomposition of an L2 space with nonlinear stochastic integrals

This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.

Interpretation: the final spatial frontier

The use of spatial econometric models in political science has steadily risen in recent years. However, the interpretation of these models has generally ignored the important substantive, and even spatial, nature of the estimated effects. This leaves many papers with a (non-spatial) interpretation of coefficients on the covariates and a brief discussion of the sign and strength of the spatial parameter. We introduce a general approach to interpreting spatial models and provide several avenues for an exposition of substantive spatial effects. Our approach can be generalized to most models in the spatial econometric taxonomy. Building on the example of the diffusion of democracy, we elucidate how our approach can be applied to modern political science problems.

Minimizing movements for oscillating energies: the critical regime

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ &gt; 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.

Wavenumber selection via spatial parameter jump

The Swift–Hohenberg equation describes an instability which forms finite-wavenumber patterns near onset. We study this equation posed with a spatial inhomogeneity; a jump-type parameter that renders the zero solution stable for x <0 and unstable for x >0. Using normal forms and spatial dynamics, we prove the existence of a family of steady-state solutions that represent a transition in space from a homogeneous state to a striped pattern state. The wavenumbers of these stripes are contained in a narrow band whose width grows linearly with the size of the jump. This represents a severe restriction from the usual constant-parameter case, where the allowed band grows with the square root of the parameter. We corroborate our predictions using numerical continuation and illustrate implications on stability of growing patterns in direct simulations. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.