The conserved phase field system with a small parameter in the n-dimensional case (n[les ]3) is considered. An asymptotic solution, describing the free interface dynamics, is constructed and justified. As the small parameter tends to zero, the limiting solution satisfies the modified Stefan problem with corrected Gibbs–Thomson law.

The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is satisfied. We obtain an explicit formula that gives the dependence of the perturbation growth rate upon the equilibrium considered and the initial data for the perturbation. Secondly, we obtain the upper and lower bounds for growing solutions of the linearized problem, and we identify the initial data corresponding to the most unstable mode (i.e. to the perturbation with maximal growth rate). All results are obtained in quite a general formulation: a nematic is inside a three-dimensional domain of an arbitrary shape and strong anchoring on the boundary is supposed; the standard equations of nematodynamics are employed as the governing equations.

We revisit the theory of filtration (slow fluid motion) through a horizontal porous stratum under the usual conditions of gently sloping fluid height profile. We start by considering the model for flooding followed by natural outflow through the endwall of the stratum, which has an explicit dipole solution as generic intermediate asymptotics. We then propose a model for forced drainage which leads to a new kind of free boundary problem for the Boussinesq equation, where the flux is prescribed as well as the height h=0 on the new free boundary. Its qualitative behaviour is described in terms of its self-similar solutions. We point out that such a class of self-similar solutions corresponds to a continuous spectrum, to be compared with the discrete spectrum of the standard Cauchy problem for the porous medium equation. This difference is due to the freedom in the choice of the flux condition allowed in our problem setting. We also consider the modifications introduced in the above models by the consideration of capillary retention of a part of the fluid. In all cases we restrict consideration to one-dimensional geometries for convenience and brevity. It is to be noted however that similar problems can be naturally posed in multi-dimensional geometries. Finally, we propose a number of related control questions, which are most relevant in the application and need a careful analysis.

Singular Systems Analysis (SSA), or time domain Principal Component Analysis (PCA), is most appropriately analysed in terms of local, moving-window spectral analysis. The behaviour of Empirical Orthogonal Functions (EOF) of this theory are examined, for continuously sampled data, in the limits of large and small window length, and for centre or end projection. Filters obtained by projecting on to these EOFs are shown to approximate local, linear band pass filters, where the EOFs depend upon the correlation structure (or the power spectral density) of the signal and the window length. Power in the spectra is not generally conserved, and projection to the endpoints of a window may not converge to the underlying signal in the absence of noise. The filters are independent of the phase of the Fourier transform, and are therefore unable to distinguish dynamically between a signal and a surrogate (phase-randomized) transform of it. Iteration of such local filters using a prediction error-based stopping criterion can and does lead to improved results, but the choice of window length must be made a priori. Hence, we introduce an iterative local filter with the window length being determined as part of the filtering procedure. This involves the determination of the predictability of the projected time series, and hence allows SSA to be used in a genuinely nonlinear way.

An initial value problem for the functional differential equation

y′(t)=Ay(t)+By(qt)+Cy′(qt)+f(t), t ≥ t0 > 0

where A, B, C are complex matrices, q∈(0, 1), and f is a vector of continuous functions, is considered in this paper. Its solution is represented in terms of the fundamental solution via the variation-of-constants formula. For some special cases, the fundamental solutions are formulated as piecewise Dirichlet series. The variation-of-constants formula is used to analysis the asymptotic behaviour of the solutions of some scalar equations, including one with variable coefficients related to coherent states of the q-oscillator algebra in quantum mechanics.