In this paper, we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrix completion problems. These guidelines are demonstrated by various illustrative examples.

The steady response of a fluid with two layers of different density in a porous medium is considered during extraction through a point sink. Supercritical withdrawal in which both layers are being withdrawn is investigated using a spectral method. We show that for each withdrawal rate, there is a single entry angle of the interface into the point sink. As the flow rate decreases the angle of entry steepens until it becomes almost vertical, at which point the method fails. This limit is shown to correspond to the upper bound on sub-critical (single-layer) flow.

We examine the dynamics of fermentation process in a yeast cell. Our investigation focuses on the main branch pathway: pyruvate and acetaldehyde branch points. We formulate the kinetics for all enzymatic reactions as Michaelis–Menten models. Since the activity of an enzyme mainly depends on the conformational changes of the enzyme structure, the enzyme requires a certain period of time to reset its structure, until it is ready to bind substrates again. For this situation, a rate-limiting step exists, for which the catalytic process suffers a delay. Since all conversion processes are catalysed by enzymes, each reaction can experience a delay at a different time. To investigate how the delay affects the reaction processes, especially at the branch points, we propose that the rate-limiting step takes place at the first reaction. For this reason, a discrete time delay is introduced to the first kinetic model. We find a bifurcation diagram for the delay that depends on the rate of glucose supply and kinetic parameters of the first enzyme. By comparison, our analysis agrees with the numerical solution. Our numerical simulations also show that there is a certain glucose supply that may optimize ethanol production.

A sporting league places every team into one of several divisions of equal size, and runs a round robin tournament for each division. Some teams are paired with another team, not necessarily in the same division, to share facilities. It is shown that however many teams are paired and whatever the pairings, it is always possible to schedule the fixtures in the minimum time, so that no two paired teams have home matches simultaneously.

Although variance swaps have become an important financial derivative to hedge against volatility risks, closed-form formulae have been developed only recently, when the realized variance is defined on discrete sample points and no continuous approximation is adopted to alleviate the mathematical difficulties associated with dealing with the discreteness of the sample data. In this paper, a new closed-form pricing formula for the value of a discretely sampled variance swap is presented under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model. With the newly found analytical formula, not only can all the hedging ratios of a variance swap be analytically derived, the numerical values of the swap price can be efficiently computed as well.

Stokes’ axisymmetrical translational motion of a slip sphere, located anywhere on the diameter of a virtual spherical fluid ‘cell’, is investigated. The fluid is micropolar and flows are parallel to the line connecting the two centres. An infinite-series solution is presented for the stream function, pressure field, vorticity, microrotation component, shear stress and couple stress of the flow. Basset-type slip boundary conditions on the sphere surface are used for velocity and microrotation. The Happel and Kuwabara boundary conditions are used on the fictitious surface of the cell model. Numerical results for the normalized drag force acting on the sphere are obtained with excellent convergence for various values of the volume fraction, the relative distance between the centre of the sphere and the virtual envelope, the vortex viscosity parameter and the slip coefficients of the sphere. In the special case when the spherical particle is in the concentric position with the cell surface, the numerical values of the normalized drag force agree with the available values in the literature.