We analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.

Standard results for boundary value problems involving second-order ordinary differential equations ensure that the existence of a well-ordered pair of lower and upper solutions together with a Nagumo condition imply existence of a solution. In this note we introduce some examples which show that existence is not guaranteed if no Nagumo condition is satisfied.

A reaction-diffusion equation with non-constant diffusivity,

ut = (D(x, t)ux)x + F(u),

is studied for D(x, t) a continuous function. The conditions under which the equation can be reduced to an equivalent constant diffusion equation are derived. Some exact forms for D(x, t) are given. For D(x, t) a stochastic function, an explicit finite difference method is used to numerically determine the effect of randomness in D(x, t) upon the speed of the reaction wave solution to Fisher's equation. The extension to two spatial dimensions is considered.

A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.

A continuous one-parameter set of binary operators on L2(R) called fractional convolution operators and which includes those of multiplication and convolution as particular cases is constructed by means of the Condon-Bargmann fractional Fourier transform. A fractional convolution theorem generalizes the standard Fourier convolution theorems and a fractional unit distribution generalizes the unit and delta distributions. Some explicit double-integral formulas for the fractional convolution between two functions are given and the induced operation between their corresponding Wigner distributions is found.

A spectral method is used to consider the porous medium combustion in an infinite slab. The infinite system of ordinary differential equations for the amplitude functions is truncated and comparisons are made for different numbers of modes included in the numerical computation. It is shown that the qualitative behaviour of the solution is captured by the first eigenmode. Dependence of the solution on initial data and a parameter is also considered.

Applying Ekeland's variational principle in this paper, we obtain a maximum principle for optimal control for a class of two-point boundary value controlled systems. The control domain need not be convex. For a special case, that is the so called LQ-type problem, we obtain the optimal control in the closed loop form and a corresponding Riccati type differential equation.

Knowledge about the foliage angle density g(α) of the leaves in the canopy of trees is crucial in foresty mangement, modelling canopy reflectance, and environmental monitoring. It is usually determined from observations of the contact frequency f(β) by solving a version of the first kind Fredholm integral equation derived by Reeve (Appendix in Warren Wilson [22]). However, for inference purposes, the practitioner uses functionals defined on g(α), such as the leaf area index F, rather than g(α) itself. Miller [12] has shown that F can be computed directly from f(β) without solving the integral equation. In this paper, we show that his result is a special case of a general transformation for linear functionals defined on g(α). The key is the existence of an alternative inversion formula for the integral equation to that derived by Miller [11].

In this paper the scattered progressive waves are determined due to progressive waves incident normally on certain types of partially immersed and completely submerged vertical porous barriers in water of infinite depth. The forms are approximate only, and are obtained using perturbation theory for nearly hard or soft barriers having high and low porosities respectively. The results for arbitrary porosity are difficult to obtain, in contrast to the well known hard limit of impermeable barriers.

Asymptotic and numerical analysis provides essential insight into the behaviour of Fourier integral solutions for the deflexion of an infinite continuously-supported flexible plate due to moving load. Thus we can define in detail how the plate deflexion depends upon the load speed, including (a) the wave patterns generated by a load moving steadily at various supercritical speeds; and (b) the time-dependent behaviour of the deflexion due to an impulsively-started load, where the two-dimensional response tends to a steady state except at the critical speed, when it grows continuously with time (in the absence of dissipation).

A local existence and uniqueness result is proved for the three-dimensional Euler-Poisson system without a pressure term which arises in plasma physics.

Derivative-type ascent formulas are deduced for the kernels of certain half-space Dirichlet problems. These have the character of differentiation formulas for the Bessel functions but involve modifying variables after completing the differentiations. The Laplace equation and the equation of generalized axially-symmetric potential theory (GASPT) are considered in these. The methods employed also permit treating abstract versions of Dirichlet problems.

Recently, considerable interest has been shown in the connection between smoothing splines and a particular class of stochastic processes. Here the connection with an equivalent class of least squares problems is used to develop algorithms, and properties of the solution are examined. We give an estimate of the condition number of the solution process and compare this with an estimate for the condition number of the Reinsch algorithm in its conventional implementation.

Gamma-Weibull variates with five parameters are defined by multiplication of gamma and Weibull densities and renormalising. Sums of independent such variates are distributed as combinations of products of gammas and confluent hypergeometric functions and are explicitly determined. Sums of independent non-identical Weibulls arise as a special case. These variates can be used to model moderately extreme scenarios between gamma and Weibull that occur in many natural applications. All results are exact.

In this paper, we investigate minimal (weak) approximate Hessians, and completely answer the open questions raised by V. Jeyakumar and X. Q. Yang. As applications, we first give a generalised Taylor's expansion in terms of a minimal weak approximate Hessian. Then we characterise the convexity of a continuously Gâteaux differentiable function. Finally some necessary and sufficient optimality conditions are presented.

In a recent paper Weber et al. [9] examined the propagation of combustion waves in a semi-infinite gaseous or solid medium. Whereas their main concern was the behaviour of waves once they had been initiated, we concentrate here on the initiation of such waves in a solid medium and have not examined in detail the steadiness or otherwise of the waves subsequent to their formation. The investigation includes calculations for finite systems. The results for a slab, cylinder and sphere are compared.

Critical conditions for initiation of ignition by a power source are established. For a slab the energy input is spread uniformly over one boundary surface. In the case of cylindrical or spherical symmetry it originates from a cylindrical core or from a small, central sphere, respectively. The size of source and reactant body is important in the last two cases. With the exception of the initial temperature distribution, the equations investigated are similar in form to those of Weber et al. [5,9] and, as a prelude to the present study, with very simple adaptation, it has been possible to reproduce the results of the earlier work. We then go on to report the result of calculations for the initiation of ignition under different geometries with various initial and boundary conditions.

In this paper we have evaluated an infinite integral of product of the Lommel and Bessel functions and powers. Some special cases of the result are discussed.