<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">1</Emphasis></Superscript> Stability of Multi-Dimensional Discrete-Velocity Boltzmann Equations

We study the L ¹ stability of multi-dimensional discrete-velocity Boltzmann equations. Under suitable smallness assumption on initial data, we show that bounded mild solutions are L ¹ stable. For a stability estimate, we employ Bonys multi-dimensional analysis for total interactions over characteristic planes.     

On the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> Theory of Hydrostatic Euler Equations

In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H ^s solutions under the local Rayleigh condition. This extends Brenier’s (Nonlinearity 12(3):495–512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak–strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H ^s a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.     

Solutions for Quasilinear Nonsmooth Evolution Systems in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>

We prove that nonsmooth quasilinear parabolic systems admit a local solution in L ^p strongly differentiable with respect to time over a bounded three-dimensional polyhedral space domain. The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems for anisotropic materials. These results are based on sharp pointwise estimates for Greens function, which are also of independent interest. To treat the nonlinear problem, we then apply a classical theorem of Sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results. As applications we have in mind primarily reaction-diffusion systems. The treatment of such equations in an L ^p context seems to be new and allows (by Gauss theorem) the proper definition of the normal component of currents across the boundary.     

Three solutions to inequalities of Dirichlet problem driven by p（x）-Laplacian

A class of nonlinear elliptic problems driven by p（x）-Laplacian with a nonsmooth locally Lipschitz potential is considered.By applying the version of the nonsmooth three-critical-point theorem,the existence of three solutions to the problems is proved.     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.     

Adhesion Energy of <Emphasis Type="Italic">C. elegans</Emphasis>

The surface adhesion between C. elegans and the agar plates on which they are commonly grown has yet to be accurately quantified. C. elegans is a scientifically important species of nematode whose simple structure allowed the first mapping of the complete nervous system in a multicellular organism. One of the current topics of research in the C. elegans community is the investigation of neuronal function in locomotion. Models of locomotion are used in these studies to aid in determination of the functions of specific neurons involved in locomotion. The adhesion force plays a critical role in developing these models. This paper presents the experimental determination of the adhesion energy of a representative sample of C. elegans. Adhesion energy was determined by a direct pull-off technique. In this approach, nematodes are anesthetized to prevent movement and secured to a small load cell before an agar plate is slowly brought into contact with the specimen and then removed. The maximum tensile force is then fit to a JKR-type adhesion model, which assumes that the nematode is a cylinder in order to determine the adhesion energy. Repeated adhesions are also investigated to determine the importance of drying on the measured adhesion force. From these experiments, the adhesion energy was found to be W =?4.94 ± 1.19 mJ/m². Limited experiments on the rol-6 cuticle mutant found a lower adhesion energy W =?2.65 ± 1.16 mJ/m² for these animals.     

The effect of 1,3:2,4-<Emphasis Type="Italic">bis</Emphasis>-<Emphasis Type="Italic">O</Emphasis>-(<Emphasis Type="Italic">p</Emphasis>-methylbenzylidene)-<Emphasis Type="SmallCaps">d</Emphasis>-sorbitol (PDTS) on uniaxial elongational viscosity of polypropylene

We investigated the dynamic viscoelasticity and elongational viscosity of polypropylene (PP) containing 0.5 wt% of 1,3:2,4-bis-O-(p-methylbenzylidene)-d-sorbitol (PDTS). The PP/PDTS system exhibited a sol–gel transition (T _gel) at 193 °C. The critical exponent n was nearly equal to 2/3, in agreement with the value predicted by a percolation theory. This critical gel is due to a three-dimensional network structure of PDTS crystals. The elongational viscosity behavior of neat PP followed the linear viscosity growth function 3η ⁺ (t), where η ⁺ (t) is the shear stress growth function in the linear viscoelastic region. The elongational viscosity of the PP/PDTS system also followed the 3η ⁺ (t) above T _gel but did not follow the 3η ⁺ (t) and exhibited strong strain-softening behavior below T _gel. This strain softening can be attributed to breakage of the network structure of PDTS with a critical stress (σ _c) of about 10⁴ Pa.     

Numerical Simulation of Nanoindentation and Patch Clamp Experiments on Mechanosensitive Channels of Large Conductance in <Emphasis Type="Italic">Escherichia coli</Emphasis>

A hierarchical simulation framework that integrates information from all-atom simulations into a finite element model at the continuum level is established to study the mechanical response of a mechanosensitive channel of large conductance (MscL) in bacteria Escherichia coli (E. coli) embedded in a vesicle formed by the dipalmitoylphosphatidycholine (DPPC) lipid bilayer. Sufficient structural details of the protein are built into the continuum model, with key parameters and material properties derived from molecular mechanics simulations. The multi-scale framework is used to analyze the gating of MscL when the lipid vesicle is subjective to nanoindentation and patch clamp experiments, and the detailed structural transitions of the protein are obtained explicitly as a function of external load; it is currently impossible to derive such information based solely on all-atom simulations. The gating pathways of E. coli-MscL qualitatively agree with results from previous patch clamp experiments. The gating mechanisms under complex indentation-induced deformation are also predicted. This versatile hierarchical multi-scale framework may be further extended to study the mechanical behaviors of cells and biomolecules, as well as to guide and stimulate biomechanics experiments.     

Mechanical properties and structure of Haliotis discus hannai Ino and Hemifusus tuba conch shells： a comparative study

Haliotis discus hannai Ino （abalone shell） and Hemifusus tuba conch shell have been studied for the purpose to comparatively investigate the mechanisms by which nature designs composites. It is shown that both shells are composed of aragonite and a small amount of proteins while the conch shell shows finer microstructure but lower strength than aba- lone shell. It is also shown that the fresh shells exhibits better property than those after heat-treatments. It is therefore sup- posed that the size of inorganic substance is not a dominant factor to improve strength, while both proteins in shells and the microstructure of inorganic matter also play important roles.     

Positive solutions to （n-1,1） m-point boundary value problems with dependence on the first order derivative

Using the extension of Krasnoselskii＇s fixed point theorem in a cone, we prove the existence of at least one positive solution to the nonlinear nth order m-point boundary value problem with dependence on the first order derivative. The associated Green＇s function for the nth order m-point boundary value problem is given, and growth conditions are imposed on the nonlinear term f which ensures the existence of at least one positive solution. A simple example is presented to illustrate applications of the obtained results.     

<Emphasis Type="Italic">In-situ</Emphasis> Measurement of Crystalline Lattice Strains in Polytetrafluoroethylene

Strain measurements by neutron diffraction are employed as an in situ technique to obtain insight into the deformation modes of crystalline domains in a deformed semi-crystalline polymer. The SMARTS (Spectrometer for MAterials Research at Temperature and Stress) diffractometer has been used to measure the crystalline lattice displacements in polytetrafluoroethylene (PTFE) for crystalline phase IV (at room temperature) in tension and compression and for crystalline phase I (at 60°C) in compression. The chemical structure of PTFE, -(C₂F₄)-_n, makes it ideally suited for investigation by neutron methods as it is free of hydrogen that results in limited penetration depths and poor diffraction acquisition in most polymers. Deformation parallel to the prismatic plane normals is shown to occur by inter-polymer chain compression with a modulus ∼10× bulk, while deformation parallel to the basal plane normal occurs by intra-polymer chain compression with a modulus ∼1000× bulk, corresponding with theoretical values for a PTFE chain modulus. Deformation parallel to the pyramidal plane normals is accommodated by inter-polymer chain shear.     

On <Emphasis Type="Italic">n</Emphasis>th-order nonlinear ordinary random differential equations

An existence result for a nonlinear nth-order ordinary random differential equation is proved under the Carathéodory condition. Two existence results for extremal random solutions are also proved in the Carathéodory case and the discontinuous case of the nonlinearity involved in the equations. Our investigations are carried out in the Banach space of continuous real-valued functions on closed bounded intervals of the real line together with the application of a random version of the Leray–Schauder principle.     

Holographic microrheology of polysaccharides from <Emphasis Type="Italic">Streptococcus mutans</Emphasis> biofilms

We use three-dimensional holographic particle tracking to perform microrheological measurements of model gelled media, including the polysaccharide pellicle of dental biofilms created by the common cariogenic oral pathogen Streptococcus mutans. Nanometer-resolution video-rate holographic tracking of embedded colloidal spheres provides accurate measurements of the gels’ complex viscoelastic moduli, including insights into these properties’ heterogeneity. When applied to polysaccharides of S. mutans biofilms, these techniques promise quantitative microscopic assays for candidate therapeutic agents against cariogenic dental biofilms.     

Study of morphological hysteresis in <Emphasis Type="Italic">partially immiscible</Emphasis> polymers

The morphology evolution of two systems of partially immiscible polymers, differing in miscibility, is investigated by means of rheological experiments and optical microscopy. For each system, two concentrations, 10% and 20%, are used. For immiscible systems, a hysteresis zone, defined by coalescence and breakup, exists where the average drop radius is not a unique function of the shear rate. We investigate whether the findings also apply to partially immiscible polymers. The average radii at different shear rates, measured with rheology, are compared to model predictions. The hysteresis zone, if present, is indeed affected by the polymeric system, the concentration and the flow history applied. Coalescence evolution is measured for three different step-downs in shear rate. For both 10% systems, the resulting average radii show a rather high scattering and do not match the theoretical predictions. For the 20% concentrations, the average experimental drop sizes seem independent of the magnitude of the step-down, at least during a certain period of time. Thereafter, it experiences a sudden, in the time scale of the experiments unbounded, increase in size that is more pronounced for the higher step-downs. Deviations of the experimental data from theoretical predictions are attributed to the partially immiscible character of the systems, yielding enhanced coalescence which, in turn, can induce confinement effects.     

Rheological and micro structural studies of methylcellulose solutions in <Emphasis Type="Italic">N</Emphasis>,<Emphasis Type="Italic">N</Emphasis>-dimethylformamide for preparation of extruded beads: controlled urea release

The rheological properties of methylcellulose in N,N-dimethylformamide (MC-DMF) gel are investigated to prepare extruded beads. The temperature scan under dynamic compression for various concentrations of MC in DMF is performed to investigate the rapture of MC gel at a constant frequency of 1 Hz. The morphological studies are performed using a scanning electron microscope (SEM) to analyze the size and shape of dried bead. However, during swelling studies, the MC beads have the capability to swell and retain a large amount of water >?9150% by weight and 9192.6% by volume. The mechanism of swelling is thermodynamically verified, where the enthalpy of hydration of initial layer of MC bead is negative. The newly defined electrostatic penta-pole model explains the anomalous behavior of urea release, where urea is assumed to be electrostatically bounded with the MC molecules.

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Approximation of Flat <Emphasis Type="Italic">W</Emphasis><Superscript>2,2</Superscript> Isometric Immersions by Smooth Ones

Let \({S\subset\mathbb{R}^2}\) be a bounded Lipschitz domain and denote by \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\) the set of mappings \({u\in W^{2,2}(S;\mathbb{R}^3)}\) which satisfy \({(\nabla u)^T(\nabla u) = Id}\) almost everywhere. Under an additional regularity condition on the boundary \({\partial S}\) (which is satisfied if \({\partial S}\) is piecewise continuously differentiable), we prove that the strong W ^2,2 closure of \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)\cap C^{\infty}(\overline{S};\mathbb{R}^3)}\) agrees with \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation

We deal with one dimensional p-Laplace equation of the form

$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$

under Dirichlet boundary condition, where \(p>2\) and \(f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.