Entropy production in open volume-preserving systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.     

Finite element analysis of mode III steady crack growth in anisotropic hardening materials

In this paper, the steady crack growth of mode III under small scale yielding conditions is investigated for anisotropic hardening materials by the finite element method. The elastic-plastic stiffness matrix for anisotropic materials is given. The results show the significant influences of anisotropic hardening behaviour on the shape and size of plastic zone and deformation field near the crack tip. With a COD fracture criterion, the ratio of stress intensity factorsk _ss/k_c varies appreciably with the anisotropic hardening parameterM and the hardening exponentN.     

Continuation of Invariant Subspaces via the Recursive Projection Method

The Recursive Projection Method is a technique for continuation of both the steady states and the dominant invariant subspaces. In this paper a modified version of the RPM called projected RPM is proposed. The modification underlines the stabilization effect. In order to improve the poor update of the unstable invariant subspace we have applied subspace iterations preconditioned by Cayley transform. A statement concerning the local convergence of the resulting method is proved. Results of numerical tests are presented.     

Convergence rates for steady-state derivative estimators

We describe various derivative estimators for the case of steady-state performance measures and obtain the order of their convergence rates. These estimatorsdo not use explicitly the regenerative structure of the system. Estimators based on infinitesimal perturbation analysis, likelihood ratios, and different kinds of finite-differences are examined. The theoretical results are illustrated via numerical examples.     

Fast volumetric spatial-spectral MR imaging of hyperpolarized C-labeled compounds using multiple echo 3D bSSFP

Purpose

The goal of this work was to develop a fast 3D chemical shift imaging technique for the noninvasive measurement of hyperpolarized ¹³C-labeled substrates and metabolic products at low concentration.

Materials and Methods

Multiple echo 3D balanced steady state magnetic resonance imaging (ME-3DbSSFP) was performed in vitro on a syringe containing hyperpolarized [1,3,3-2H3; 1-¹³C]2-hydroxyethylpropionate (HEP) adjacent to a ¹³C-enriched acetate phantom, and in vivo on a rat before and after intravenous injection of hyperpolarized HEP at 1.5 T. Chemical shift images of the hyperpolarized HEP were derived from the multiple echo data by Fourier transformation along the echoes on a voxel by voxel basis for each slice of the 3D data set.

Results

ME-3DbSSFP imaging was able to provide chemical shift images of hyperpolarized HEP in vitro, and in a rat with isotropic 7-mm spatial resolution, 93 Hz spectral resolution and 16-s temporal resolution for a period greater than 45 s.

Conclusion

Multiple echo 3D bSSFP imaging can provide chemical shift images of hyperpolarized ¹³C-labeled compounds in vivo with relatively high spatial resolution and moderate spectral resolution. The increased signal-to-noise ratio of this 3D technique will enable the detection of hyperpolarized ¹³C-labeled metabolites at lower concentrations as compared to a 2D technique.     

Spatiotemporal complexity of a diffusive planktonic system with prey-taxis and toxic effects

In this paper, we propose a three-species reaction-diffusion planktonic system with prey-taxis and toxic effects, in which the zooplankton can recognize the nontoxic and toxin-producing phytoplankton and can make proper response. We first establish the existence and stability of the unique positive constant equilibrium solution by utilizing the linear stability theory for partial differential equations. Then we obtain the existence and properties of nonconstant positive solutions by detailed steady state bifurcation analysis. In addition, we obtain that change of taxis rate will result in the appearance of time-periodic solutions. Finally, we conduct some numerical simulations and give the conclusions.     

Existence of the solutions of a reaction cross-diffusion model for two species

In this paper, we investigate a two-cooperative species model with reaction cross-diffusion under the Dirichlet boundary value condition. By applying Lyapunov–Schmidt reduction method and some important formulas, we obtain the existence of coexistence solutions under some given conditions.     

Analysis of a prey-predator model with disease in prey

In this paper, a system of reaction-diffusion equations arising in eco-epidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the predator is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.     

Complete set of periodic solutions of a discontinuous recurrence equation

A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang‐bang control is studied. We show that each solution is periodic and its prime period can be determined by three of its consecutive terms. By means of this periodicity property, we may then solve the steady state problem which to our knowledge is not solved by other means.