The structure of space–time is examined by extending the standard Lorentz connection group to its complex covering group, operating on a 16-dimensional “spinor” frame. A Hamiltonian variation principle is used to derive the field equations for the spinor connection. The result is a complete set of field equations which allow the sources of the gravitational and electromagnetic fields, and the intrinsic spin of a particle, to appear as a manifestation of the space–time structure. A cosmological solution and a simple particle solution are examined. Further extensions to the connection group are proposed.

We consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.

Given a set of points in the plane, the problem of existence and finding the least absolute deviations line is considered. The most important properties are stated and proved and two efficient methods for finding the best least absolute deviations line are proposed. Compared to other known methods, our proposed methods proved to be considerably more efficient.

An analysis is developed for the behaviour of a cloud of cavitation bubbles during both the growth and collapse phases. The theory is based on a multipole method exploiting a modified variational principle developed by Miles [“Nonlinear surface waves in closed basins”, J. Fluid Mech.75 (1976) 418–448] for water waves. Calculations record that bubbles grow approximately spherically, but that a staggered collapse ensues, with the outermost bubbles in the cloud collapsing first of all, leading to a cascade of bubble collapses with very high pressures developed near the cloud centroid. A more complex phenomenon occurs for bubbles of variable radius with local zones of collapse, with a complex frequency spectrum associated with each individual bubble, leading to both local and global collective behaviour.

This paper studies the problem of delay-dependent robust H∞ control for singular systems with multiple delays. Based on a Lyapunov–Krasovskii functional approach, an improved delay-dependent bounded real lemma (BRL) for singular time-delay systems is established without using any of the model transformations and bounding techniques on the cross product terms. Then, by applying the obtained BRL, a delay-dependent condition for the existence of a robust state feedback controller, which guarantees that the closed-loop system is regular, impulse free, robustly stable and satisfies a prescribed H∞ performance index, is proposed in terms of a nonlinear matrix inequality. The explicit expression for the H∞ controller is designed by using linear matrix inequalities and the cone complementarity iterative linearization algorithm. Numerical examples are also given to illustrate the effectiveness of the proposed method.

The stability characteristics of an infinite horizontal fluid layer excited by a time-periodic, sinusoidally varying free-boundary temperature, have been investigated numerically using the Floquet theory. It has been found that the modulation of the temperature gradient across the fluid layer affects the onset of the Rayleigh–Bénard convection. Modulation can give rise to instability in the subcritical conditions and it can also suppress the instability in the supercritical cases. The instability in the fluid layer manifests itself in the form of either a harmonic or subharmonic flow, controlled by thermal modulation.

In two recent papers a global upper bound is derived for Jensen’s inequality for weighted finite sums. In this paper we generalize this result on positive normalized functionals.

In this paper we extend results of Inoan and Kolumban on pseudomonotone set-valued mappings to topological vector spaces. An application is made to a variational inequality problem.

There is increasing interest in the cost of railway infrastructure access as a variety of private companies operate trains with different lengths over a common rail network. It is important to have a method for evaluating the cost of adding an additional train to a railway timetable. This is particularly so for single line track with occasional passing loops. The concept of a fixed loop capacity is extended to one that is dependent on the trains. We develop a model for scheduling a heterogeneous set of trains on single line systems with loops. Our method minimizes the total weighted delay. A Lagrangian relaxation technique is used that relaxes the capacity constraints for track segments and super segments. We measure the delay for each train and the total weighted delay for the heterogeneous set of trains. Our model allows us to investigate the robustness of the weighted delay to variation in the departure time of individual trains. The paper demonstrates that a Lagrangian relaxation heuristic provides optimal train schedules for instances of small heterogeneous train sets. The method is used primarily to check the effectiveness of heuristic algorithms commonly used to find schedules for practical problems.