A local description of space and time in which translations are included in the group of gauge transformations is studied using the formalism of fibre bundles. It is shown that the flat Minkowski space–time may be obtained from a non-flat connection in a de Sitter structured fibre bundle by choosing at least two different cross-sections. The interaction terms in the covariant derivative of a Dirac wave function that correspond to translations may be interpreted as the mass term of the Dirac equation, and then the two cross-sections (gauges) correspond to the description of a fermion and antifermion respectively.

An axially symmetric metric in oblate spheroidal co-ordinates is considered. Two exact solutions of the field equations corresponding to zero mass meson fields are obtained. The details of the solutions are also discussed. These solutions are also generalized to include electromagnetic fields.

We discuss modelling and simulation of volumetric rainfall in a catchment of the Murray–Darling Basin – an important food production region in Australia that was seriously affected by a recent prolonged drought. Consequently, there has been sustained interest in development of improved water management policies. In order to model accumulated volumetric catchment rainfall over a fixed time period, it is necessary to sum weighted rainfall depths at representative sites within each sub-catchment. Since sub-catchment rainfall may be highly correlated, the use of a Gamma distribution to model rainfall at each site means that catchment rainfall is expressed as a sum of correlated Gamma random variables. We compare four different models and conclude that a joint probability distribution for catchment rainfall constructed by using a copula of maximum entropy is the most effective.

The machinery of Lie theory (groups and algebras) is applied to the unsteady equations of motion of rotating fluid. A special-function type solution for the steady state is derived. It is then shown how the solution generates an infinite number of time-dependent solutions via three arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

We study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.

We define an integral function Iμ(α, x; a, b) for non-negative integral values of μ by

It is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

The acoustic response of a two-dimensional nearly-closed cavity to an excitation through a small opening is examined, using the method of matched asymptotic expansions. The Helmholtz mode of vibration is discussed using a low-frequency expansion of the velocity potential in the cavity interior. The variation in frequency and magnitude of the resonator response is explored, both for the Helmholtz and the natural-frequency modes.

The diffusion equation is used to model and analyze sorption, a process used in the purification or separation of fluids. For the diffusion inside a spherical porous solid immersed in a limited-volume and well-stirred fluid, Ruthven [5], Crank [3] and, for the analogous flow of heat, Carslaw and Jaeger [2] give an eigenfunction expansion solution to the diffusion equation that provides accurate long-time solutions when only a few terms are used. However, to obtain the same accuracy for short-time solutions the number of eigenfunction terms required increases exponentially. An alternative error function solution of Carman and Haul [1] is accurate for sufficiently short times but not for long times. Although their solution is well quoted [3, 4, 6], Carman and Haul do not provide a derivation in their paper. This paper provides a full derivation of the short-time solution of Carman and Haul that uses only the first term of a negative exponential series in the Laplace domain. It is shown that the accuracy and range of the short-time result is improved by the inclusion of additional terms of the negative exponential series. An analysis of short-time and long-time resultsis presented, together with recommendations as to their use.

In this paper a control problem with a cost functional depending on the number of switchings and on the speed of alterations of control is considered. Necessary conditions for the existence of an optimal solution are given.

In this paper we consider a natural extension of the minimum time problem in optimal control theory which we refer to as the minimum trapping time problem. The minimum trapping time problem requires a fixed time interval [0, T], where T is finite. The aim is to determine a control for which the system trajectory not only reaches a specified target in minimum time but also remains trapped within the target until time T. Our aim is to devise a computational procedure for solving the minimum trapping time problem. The computational procedure we adopt uses control parametrisation in which the class of controls is approximated by a class of piecewise constant functions. The problem we are solving is therefore an approximation to the original minimum trapping time problem. Some properties for the approximate problem are then established. These lead to an extremely efficient iterative procedure for calculating the minimum trapping time.

In this paper we resolve the problem of controllability of nonlinear interconnected systems of neutral type. We consider two types of systems, a general one, and one in which some control appears linearly. In each case we insist that each isolated system of the interconnected problem is controlled by its own variables while taking into account the interacting effects. Controllability is proved by assuming some controllability criteria of each isolated system and some growth condition of the interconnecting function. Fixed point and open mapping theorems are used. Examples from economics and engineering are presented.

For the cubic Schrödinger equation iut = uxx + k|u|2u, 0 ≤ x, t < ∞, initial data u(x, 0) = u0(x) ∈ H2[0, ∞), and Robin boundary data ux(0, t) + αu(0, t) = R(t) ∈ C2[0, ∞) (where α is real), we show that the solution u depends continuously on u0 and R.

For diffusion problems, the boundary conditions are specified at two distinct points, yielding a two end-point boundary value problem which normally requires iterative techniques. For spherical geometry, it is possible to specify the boundary conditions at the same points, approximately, by using an optimization principle for arbitrary diffusivity. When the diffusivity obeys a power or an exponential law, a first integral exists and iteration can be avoided. For those two exact cases, it is shown that the general optimization result is extremely accurate when diffusivity increases rapidly with concentration.

Temperature distributions recorded by thermocouples in a solid body (slab) subject to surface heating are used in a mathematical model of two-dimensional heat conduction. The corresponding Dirichlet problem for a holomorphic function (complex potential), involving temperature and a heat stream function, is solved in a strip. The Zhukovskii function is reconstructed through singular integrals, involving an auxiliary complex variable. The complex potential is mapped onto an auxiliary half-plane. The flow net (orthogonal isotherms and heat lines) of heat conduction is compared with the known Carslaw–Jaeger solution and shows a puzzling topology of three regimes of energy fluxes for temperature boundary conditions common in passive thermal insulation. The simplest regime is realized if cooling of a shaded zone is mild and heat flows in a slightly distorted “resistor model” flow tube. The second regime emerges when cooling is stronger and two disconnected separatrices demarcate the back-flow of heat from a relatively hot segment of the slab surface to the atmosphere through relatively cold parts of this surface. The third topological regime is characterized by a single separatrix with a critical point inside the slab, where the thermal gradient is nil. In this regime the back-suction of heat into the atmosphere is most intensive. The closed-form solutions obtained can be used in assessment of efficiency of thermal protection of buildings.

We develop a computational method for solving an optimal control problem governed by a switched impulsive dynamical system with time delay. At each time instant, only one subsystem is active. We propose a computational method for solving this optimal control problem where the time spent by the state in each subsystem is treated as a new parameter. These parameters and the jump strengths of the impulses are decision parameters to be optimized. The gradient formula of the cost function is derived in terms of solving a number of delay differential equations forward in time. Based on this, the optimal control problem can be solved as an optimization problem.

We propose a new primal-dual interior-point algorithm based on a new kernel function for linear optimization problems. New search directions and proximity functions are proposed based on the kernel function. We show that the new algorithm has and iteration bounds for large-update and small-update methods, respectively, which are currently the best known bounds for such methods.

We explore the solvability of a general system of nonlinear relaxed cocoercive variational inequality (SNVI) problems based on a new projection system for the direct product of two nonempty closed and convex subsets of real Hilbert spaces.

Recently, Srivastava, Pathan and Kamarujjama established several results for generalised Voigt functions which play an important role in several diverse fields of physics—such as astrophysical spectroscopy and the theory of neutron reactions. In the present paper we aim to generalise some partly bilateral and partly unilateral representations and generating functions of Srivastava et al. by considering a specialised version of the Srivastava-Chen definition of the unified Voigt functions. Several special cases of our main results are mentioned briefly.

We study the problem of pseudostate and static output feedback stabilization for singular fractional-order linear systems with fractional order $\unicode[STIX]{x1D6FC}$ when $0<\unicode[STIX]{x1D6FC}<1$. All the results are given by linear matrix inequalities. First, a new sufficient and necessary condition for the admissibility of singular fractional-order systems is presented. Then based on the admissible result, not only are sufficient conditions for designing pseudostate and static output feedback controllers obtained, but also sufficient and necessary conditions are presented by using different methods that guarantee the admissibility of the closed-loop systems. Finally, the effectiveness of the proposed approach is demonstrated by numerical simulations and a real-world example.

The vertical rise of a round plume of light fluid through a surrounding heavier fluid is considered. An inviscid model is analysed in which the boundary of the plume is taken to be a sharp interface. An efficient spectral method is used to solve this nonlinear free-boundary problem, and shows that the plume narrows as it rises. A generalized condition is also introduced at the boundary, and allows the ambient fluid to be entrained into the rising plume. In this case, the fluid plume first narrows then widens as it rises. These features are confirmed by an asymptotic analysis. A viscous model of the same situation is also proposed, based on a Boussinesq approximation. It qualitatively confirms the widening of the plume due to entrainment of the ambient fluid, but also shows the presence of vortex rings around the interface of the rising plume.