Tissues form from collections of cells that interact together mechanically via cell-to-cell adhesion, mediated by transmembrane cell adhesion molecules. Under a sufficiently large amount of induced stress, these tissues can undergo elastic deformation in the direction of tension, where they then elongate without any topological changes, and experience plastic deformation within the tissue. In this work, we present a novel mathematical model describing the deformation of cells, where tissues are elongated in a controlled manner. In doing so, the cells are able to undergo remodelling through elastic and then plastic deformation, in accordance with experimental observation. Our model describes bistable sizes of a cell that actively deform under stress to elongate the cell. In the absence of remodelling, the model reduces to the standard linear interaction model. In the presence of instant remodelling, we provide a bifurcation analysis to describe the existence of the bistable cell sizes. In the case of general remodelling, we show numerically that cells within a tissue may populate both the initial and elongated cell sizes, following a sufficiently large degree of stress.

A viscous fluid is confined between two smooth horizontal walls, in a vertical channel. The upper wall may move with constant speed, but the lower wall is stationary and a portion of it is heated. A plume of heated fluid develops, and may also be swept downstream by the motion of the upper wall. When the heating effect is small and the upper plate does not move, a closed-form solution for the temperature profile is presented. A numerical spectral method is then presented, and allows highly accurate nonlinear solutions to be obtained, for the temperature and the fluid motion. These are compared against the closed-form solution in the linearized case, and the effects of nonlinearity on temperature and velocity are revealed. The results also show that periodic plume shedding from the heated region can occur in the nonlinear case.

The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.

Double-masking may be used to reduce the transmission of a virus. If additionally the masks are compressible, with different permeabilities and behaviour under compression, then it may be possible to design a mask that allows for easy breathing under normal breathing conditions, but is relatively impermeable under coughing or sneezing conditions. Such a mask could be both comfortable to wear and effective. We obtain analytical solutions for the steady-state flow-through behaviour of such a double mask under flow-out conditions. The results show that the reduction in permeability required to produce a relatively impermeable mask under high flux expulsion (sneezing) conditions could be achieved using either a single filter compressible mask or two filters with different poroelastic parameters. The parameters can be more easily adjusted using a double mask. For both single- and double-mask cases, there is an abrupt cut off, whereby through-flux levels reduce from a maximum value to zero as pressure drop levels increase beyond a critical value. Additionally, in the double-mask case, there exists a second steady-state solution for particular parameter ranges. This second solution is unlikely to occur under normal circumstances.

This paper examines the issue of derivative pricing within the framework of a fractional stochastic volatility model. We present a deterministic partial differential equation system to derive an approximate expression for the derivative price. The proposed approach allows for the stochastic volatility to be expressed as a composition of deterministic functions of time and a fractional Ornstein–Uhlenbeck process. We apply this method to the European option pricing under the fractional Stein–Stein volatility model, demonstrating its feasibility and reliability through numerical simulations. Our numerical simulations also illustrate the impact of the parameters in the fractional stochastic volatility model on the option price.

Thin spray-on liners (TSLs) have been found to be effective for structurally supporting the walls of mining tunnels and thus reducing the occurrence of rock bursts, an effect primarily due to the penetration of cracks by the liner. Surface tension effects are thus important. However, TSLs are also used to simply stabilize rock surfaces, for example, to prevent rock fall, and in this context crack penetration is desirable but not necessary, and the tensile and shearing strength and adhesive properties of the liner determine its effectiveness. We examine the effectiveness of nonpenetrating TSLs in a global lined tunnel and in a local rock support context. In the tunnel context, we examine the effect of the liner on the stress distribution in a tunnel subjected to a geological or mining event. We show that the liner has little effect on stresses in the surrounding rock and that tensile stresses in the rock surface are transmitted across the liner, so that failure is likely to be due to liner rupture or detachment from the surface. In the local rock support context, loose rock movements are shown to be better achieved using a liner with small Young’s modulus, but high rupture strength.

A nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.

In this paper, we investigate the thermal evolution in a one-dimensional bagasse stockpile. The mathematical model involves four unknowns: the temperature, oxygen content, liquid water content and water vapour content. We first nondimensionalize the model to identify dominant terms and so simplify the system. We then calculate solutions for the approximate and full system. It is shown that under certain conditions spontaneous combustion will occur. Most importantly, we show that spontaneous combustion can be avoided by sequential building. To be specific, in a situation where, say, a $4.7\,$m stockpile can spontaneously combust, we could construct a $3\,$m pile and then some days later add another $1.7\,$m to produce a stable $4.7\,$m pile.

Based on biochemical kinetics, a stochastic model to characterize wastewater treatment plants and dynamics of river water quality under the influence of random fluctuations is proposed in this paper. This model describes the interaction between dissolved oxygen (DO) and biochemical oxygen demand (BOD), and is in the form of stochastic differential equations driven by multiplicative Gaussian noises. The stochastic persistence problem for the model of the system is analysed. Further, a numerical simulation of the stationary probability distributions of BOD and OD by approximations of the stochastic process solution is presented. These results have implications for the prediction and control of pollutants.

We provide an analytic solution of the Rössler equations based on the asymptotic limit $c\to \infty $ and we show in this limit that the solution takes the form of multiple pulses, similar to “burst” firing of neurons. We are able to derive an approximate Poincaré map for the solutions, which compares reasonably with a numerically derived map.

Sending microwaves through bauxite ore allows almost continuous measurement of moisture content during offload by conveyor belt from a ship. Data and results from a microwave analyser were brought to a European Study Group with Industry at the University of Limerick, with the over-arching question of whether the results are accurate enough. The analyser equipment uses linear regression against phase shifts and signal attenuation to infer moisture content in real time. Simple initial modelling conducted during the Study Group supports this use of linear regression for phase shift data. However, that work also revealed striking and puzzling differences between model and attenuation data.

We present an improved model that allows for multiple reflections of travelling microwaves within the bauxite and in the air above it. Our new model uses four differential equations to describe how electric fields change with distance in each of four layers. By solving these equations and taking reflections into account, we can accurately predict what the receiving antenna will pick up.

Our new solution provides much-improved matches to data from the microwave analyser, and indicates the deleterious effects of reflections. Modelled signal strength behaviour features a highly undesirable noninvertible dependence on bauxite mixture permittivity.

Practical measures that might be expected to reduce the effects of microwave reflections and improve the accuracy of microwave analyser results are suggested based on our improved model solution. This modelling approach and these results are anticipated to extend to the analysis of moisture content during transport on conveyor belts of other ores, slurries, coal, grains and pharmaceutical powders, especially when the depth of the conveyed material is variable.

To compute the maximum speed threshold for helicopters, we model the lift produced by the rotor blades. Using this model, we derive limits for each method traditionally used to alleviate dissymmetry of lift. Additionally, we find the minimum rotor angular velocity required to produce a prescribed lift at a given forward velocity. We derive conditions on the coefficient of lift for helicopter airfoils that maintain altitude. Further considerations are also made with regard to the properties of the air and its effect on helicopter dynamics.

Six patents were secured by E. H. Lanier from 1930 to 1933 for aeroplane designs that were intended to be exceptionally stable. A feature of five of these was a flow-induced “vacuum chamber” which was thought to provide superior stability and increased lift compared to typical wing designs. Initially, this chamber was in the fuselage, but later designs placed it in the wing by replacing a section of the upper skin of the wing with a series of angled slats. We report upon an investigation of the Lanier wing design using inviscid aerodynamic theory and viscous numerical simulations. This took place at the 2005 Australia–New Zealand Mathematics-in-Industry Study Group. The evidence from this investigation does not support the claims but, rather, suggests that any improvement in lift and/or stability seen in the few prototypes that were built was, most probably, due to thicker airfoils than were typical at the time.

The effects of apparatus-induced dispersion on nonuniform, density-dependent flow in a cylindrical soil column were investigated using a finite-element model. To validate the model, the results with an analytical solution and laboratory column test data were analysed. The model simulations confirmed that flow nonuniformities induced by the apparatus are dissipated within the column when the distance to the apparatus outlet exceeds $3R/2$, where R represents the radius of the cylindrical column. Furthermore, the simulations revealed that convergent flow in the vicinity of the outlet introduces additional hydrodynamic dispersion in the soil column apparatus. However, this effect is minimal in the region where the column height exceeds $3R/2$. Additionally, it is found that an increase in the solution density gradient during the solute breakthrough period led to a decrease in flow velocity, which stabilized the flow and ultimately reduced dispersive mixing. Overall, this study provides insights into the behaviour of apparatus-induced dispersion in nonuniform, density-dependent flow within a cylindrical soil column, shedding light on the dynamics and mitigation of flow nonuniformities and dispersive mixing phenomena.