Continuous dependence on data for bilocal difference equations

The continuous dependence on data is studied for a class of second order difference equations governed by a maximal monotone operator A in a Hilbert space. A nonhomogeneous term f appears in the equation and some bilocal boundary conditions a, b are added. One shows that the function which associates to {a, b, A, f} the solution of this boundary value problem is continuous in a specific sense. One uses the convergence of a sequence of operators in the sense of the resolvent. The problem studied here is the discrete variant of a problem from the continuous case.     

Multiple attractors in a Leslie–Gower competition system with Allee effects

We investigate asymptotic dynamics of the classical Leslie–Gower competition model when both competing populations are subject to Allee effects. The system may possess four interior steady states. It is proved that for certain parameter regimes both competing populations may either go extinct, coexist or one population drives the other population to extinction depending on initial conditions.     

An explicit algorithm for monotone variational inequalities

We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions.     

Two negative minima of the first normal stress difference in a cellulose‐based cholesteric liquid crystal: Helix uncoiling

The shear rate dependence of material functions such as shear viscosity (η) and the first normal stress difference (N₁) were given and interpreted earlier by Kiss and Porter. Their widely accepted work revealed the possibility of having a negative minimum of N₁ for polymeric liquid crystals. In this work, we disclose for the first time the evidence of two negative N₁ minima on a sheared cellulosic lyotropic system. The lower shear rate minimum is ascribed to the uncoiling of the cholesteric helix, as theoretically predicted earlier. Our findings contribute also to the understanding of the other minimum already reported in the literature and attributed to the nematic director tumbling mode. Moreover, the elastic change that the LC‐HPC sample undergoes during the helix unwinding of the cholesteric structure is also by means of oscillatory measurements. This study is a contribution for the understanding of the structure‐properties relationship linked with the complex rheological behavior of chiral nematic cellulose‐based systems and may help to improve their further processing. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 821–830     

SALS,WAXS and mechanical properties of heat‐treated thermotropic polymers

The influence of heat treatment on the texture, microstructure and tensile mechanical properties of extruded thin films of a series of high‐performance thermotropic liquid crystal polymers (LCPs) was investigated. LCPs based on random units of hydroxybenzoic acid (B), hydroxynaphthoic acid (N), terephthalic acid (TA) and biphenol (BP) were kindly supplied by the former Hoechst Celanese Corp as 50 µm thick extruded tapes. The LCPs, denoted B‐N, COTBP and RD1000, have B and N as common comonomers and vary the other comonomers. Thus, this study also enables the investigation of the influence of monomer composition on microstructure and mechanical properties. Heat treatments were carried out at temperatures close to the solid‐to‐nematic transition (T_s→n) for periods up to 5 h, under dry air conditions. The thermal treatment produced either two endotherms or a small increase of T_s→n (B‐N and RD1000), or increased significantly T_s→n (COTBP). Moreover, when heat treatment was carried out approximately 40°C below the respective T_s→n, the mechanical Young's modulus, E, along the extrusion axis, increased for all LCPs. Strikingly, for COTBP, E increased over 100% relative to the as‐extruded film. The results also showed that the optimum treatment time for improving the Young modulus, under dry air atmosphere, was between 3 and 4 h. Wide‐angle X‐ray scattering showed a significant sharpening of crystalline reflections and concentration of the 002 meridional reflection as a result of thermal treatment, suggesting the elimination of defects and a better alignment of the molecular chains along the extrusion axis. This would explain the increase in tensile modulus. Copyright © 2013 John Wiley & Sons, Ltd.     

Synthesis and characterization of side chain cholesteric liquid crystalline elastomers derived from chiral bisolefinic crosslinking units

In this work we prepared a nematic monomer (4′‐allyloxybiphenyl 4′‐ethoxybenzoate, M₁ ), a chiral crosslinking agent (isosorbide 4‐allyloxybenzoyl bisate, M₂ ) and a series of new side chain cholesteric liquid crystalline elastomers derived from M₁ and M₂ . The chemical structures of the monomers and polymers were confirmed by FTIR and ¹H NMR spectroscopy. The mesomorphic properties were investigated by differential scanning calorimetry, thermogravimetric analysis, polarizing optical microscopy and X‐ray diffraction. The effect of the content of the crosslinking unit on phase behaviour of the elastomers is discussed. Polymer P₁ showed a nematic phase, P₂ –P₇ showed a cholesteric phase; P₆ formed a blue Grandjean texture over a broad temperature range 145–209.6°C, with no changed on the cooling. Polymers P₄ –P₇ , with more than 6?mol?% of chiral crosslinking agent, gave rise to selective reflection. Elastomers containing less than 15?mol?% of the crosslinking units displayed elasticity, reversible phase transition with wide mesophase temperature ranges, and high thermal stability. Experimental results demonstrated that, with increasing content of crosslinking agent, the glass transition temperatures first fell and then increased; the isotropization temperatures and mesophase temperature ranges decreased.     

Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time‐asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter ε tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

On global stability of an HIV pathogenesis model with cure rate

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4⁺ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12 : 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4⁺ T cells. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non‐uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage. Copyright © 2014 John Wiley & Sons, Ltd.     

Optical Mechanotransduction with Carbazole‐Based Luminescent Liquid Single‐Crystal Elastomers

Carbazole‐based liquid single‐crystal elastomers (LSCEs) are valuable fluorescent flexible materials to perform optical mechanotransduction under ambient conditions. Indeed, the covalent incorporation of carbazole derivatives into nematic LSCEs allows to tune their luminescence on demand under mechanical control in a quick and reversible fashion. Specifically, the fluorescence intensity for these materials can be switched back and forth in less than a second. Moreover, such a process can be performed several times without detecting any sign of fatigue in the system. In addition, these materials show excellent resistance to aging; 2 years after their preparation they exhibit the very same mechanofluorescent behavior as when freshly prepared. In fact, the here reported fluorescent systems are highly sensitive; the application of a force of 70 mN decreases the fluorescence in the elastomeric material by 7%. Thus, mechanical forces are attractive external stimuli to modulate the fluorescence of nematic elastomers rapidly and reversibly enabling thereby mechanotransduction.