Ignition or thermal explosion in an oxidizing porous body of material can be described by a dimensionless reaction–diffusion equation of the form ∂tu=∇2u+λe−1/u. Here such equations will be formulated in symmetrically shaped bounded regions Ω, effectively reducing the mathematical formulation to that of one dimension. This is critically re-examined from a modern perspective using numerical methods. A computer algorithm is constructed and used to carry out a broad-ranging evaluation of the watershed critical initial temperature conditions for thermal ignition of nonuniform assemblies. It is then shown how the resulting mathematical structure for the ignition threshold curves can be correlated by a hyperbolic conic section with a high degree of accuracy over the full range of positive ambient temperature values. However, this sometimes over-predicts (which is bad) and sometimes under-predicts (which is good) the critical initial condition. The definition of additional dimensionless parameters is found to generate further simplification, leading to a universal correlating form capable of collapsing the entire solution space onto a single line in the plane of the new variables. In addition, this study considers the physically intuitive conjecture that spatial moments of the initial temperature profile ought to possess a direct mathematical link to the critical ignition threshold. As such, the mth-order spatial moment of the critical total energy content integrals is defined, and an empirical result is derived stating that certain orders of this moment should be insensitive to changes in ambient temperature and initial shape profile and may be considered functionally dependent on the dimensionless eigenvalue only, within some quantifiable error band. Spatial moment integrals, based on computed critical threshold conditions, are found to support this conjecture, with the best accuracy obtained for the second-order moments.

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.

Linear multiplicative programs are an important class of nonconvex optimisation problems that are currently the subject of considerable research as regards the development of computational algorithms. In this paper, we show that mathematical programs of this nature are, in fact, a special case of more general signomial programming, which in turn implies that research on this latter problem may be valuable in analysing and solving linear multiplicative programs. In particular, we use signomial programming duality theory to establish a dual program for a nonconvex linear multiplicative program. An interpretation of the dual variables is given.

Ronney and Sivashinsky [2] and Buckmaster and Lee [1] have proposed a certain non-autonomous first order ordinary differential equation as a simple model for an expanding spherical flame front in a zero-gravity environment. Here we supplement their preliminary numerical calculations with some analysis and further numerical work. The results show that the solutions either correspond to quenching, or to steady flame front propagation, or to rapid expansion of the flame front, depending on two control parameters. A crucial component of our analysis is the construction of a barrier orbit which divides the phase plane into two parts. The location of this barrier orbit then determines the fate of orbits in the phase plane.

Reaction-diffusion systems are widely used to model the population densities of biological species competing for natural resources in their common habitat. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishing strictly positive uniform lower bounds (when they exist) can be quite troublesome. Two previously established criteria for the permanence (non-extinction and non-explosion) of solutions of general weakly-coupled competition-diffusion systems with diagonally convex reaction terms are used here as background to develop more easily verifiable and concrete conditions for permanence in various well-known competition diffusion models. These models include multi-component reaction-diffusion systems with (i) the by now classical Lotka-Volterra (logistic) reaction terms, (ii) higher order “logistic” interaction between the species, (iii) logistic-logarithmic reaction terms, (iv) Ayala-Gilpin-Ehrenfeld θ-interaction terms (which are used to model Drosophila competition), (v) logistic-exponential interaction between the species, (vi) Schoener-exploitation and (vii) modified Schoener-interference between the species. In (i) a known condition for permanence (for the ODE-system) is recovered, while in (ii)–(vii) new criteria for permanence are established.

The paper analyes the dynamics of duopoly output game involving a warfare strategy proposed by Robert Bishop. Necessary and sufficient conditions are obtained for the stability of a duopoly warfare game.

Selecting important variables and estimating coordinate covariation have received considerable attention in the current big data deluge. Previous work shows that the gradient of the regression function, the objective function in regression and classification problems, can provide both types of information. In this paper, an algorithm to learn this gradient function is proposed for nonidentical data. Under some mild assumptions on data distribution and the model parameters, a result on its learning rate is established which provides a theoretical guarantee for using this method in dynamical gene selection and in network security for recognition of malicious online attacks.

The state of a patient is an important concept in biomedical sciences. While analytical methods for predicting and exploring treatment strategies of disease dynamics have proven to have useful applications in public health policy and planning, the state of a patient has attracted less attention, at least mathematically. As a result, models constructed in relation to treatment strategies may not be very informative. We derive a patient-dependent parameter from an age-physiology dependent population model, and show that a single treatment strategy is not always optimal. Also, we derive a function which increases with the patient dependence parameter and describes the effort expended to be in good health.

An efficient probabilistic algorithm is presented for the determination of the rate matrix of a block-GI/M/1 Markov chain. Recurrence of the chain is not assumed.

Two generalised shallow-shell bending elements are developed for the analysis of doubly-curved shallow shells having arbitrarily shaped plan-forms. Although both elements are formulated using the concept of iso-deflection contour lines, one element uses the three displacement components U, V and W as the basic unknowns, while the displacement component W and the stress function ф serve as the unknowns in the other element. A number of illustrative examples are included to demonstrate the accuracy and relative convergence of the proposed shallow-shell elements when employed for static analysis purposes.

This paper examines the role of import tariffs and consumption taxes when a product is supplied to a domestic market by a foreign monopoly via a subsidiary. It is assumed that there is no competition in the domestic market from internal suppliers. The home country is able to levy a profits tax on the subsidiary; the objective of our analysis is to determine the levels of tariff or consumption tax which maximise national welfare. Comparisons are made under the two alternative policies from the perspectives of national welfare, total national cost and average national cost. The major policy implication of the analysis is that a consumption tax is the more effective instrument for maximising national welfare provided the profits tax is less than a certain critical value; if the profits tax exceeds this value then a tariff, though in the form of a subsidy, is the most effective instrument. Our results complement, correct and extend an earlier analysis by Katrak (1977) [6].

When the first two elements of a sequence satisfying a second order difference equation are prescribed, the remaining elements are evaluated from a continued fraction and a simple product.

The inverse spectral method for a general N × N spectral problem for solving nonlinear evolution equations in one spacial and one temporal dimension is extended to include multi-boundary jumps and high-order poles and their explicit representations. It therefore provides a formalism to generate soliton solutions that correspond to higher-order poles of the spectral data.

We construct a set of functions, say, composed of a cosine function and a sigmoidal transformation γr of order r > 0. The present functions are orthonormal with respect to a proper weight function on the interval [−1, 1]. It is proven that if a function f is continuous and piecewise smooth on [−1, 1] then its series expansion based on converges uniformly to f so long as the order of the sigmoidal transformation employed is 0 < r ≤ 1. Owing to the variational feature of according to the value of r, one can expect improvement of the traditional Fourier series approximation for a function on a finite interval. Several numerical examples show the efficiency of the present series expansion in comparison with the Fourier series expansion.

Two eccentric rotating cylinders together with a permeable membrane surrounding the inner cylinder are used to model the flow around a modified viscometer. A perturbation method is used to solve for the flow between the membrane and the outer cylinder; the flow between the inner rotor and the membrane is assumed to be governed by Stoke's equation, and the two flow regimes are coupled by the through-flow across the membrane. For moderate values of Reynolds number and eccentricity, the permeability of the membrane plays a negligible role, and the flow through the membrane is found to be eccentricity dependent. High eccentricities result in the formation of eddies which, upon increasing the Reynolds number, move in a direction opposite to that of the rotation of the outer bowl.

In this paper, multigrid methods for solving the biharmonic equation using some nonconforming plate elements are considered. An average algorithm is applied to define the transfer operator. A general analysis of convergence is given.

This work deals with low-frequency asymptotic solutions using the method of matched asymptotic expansions. It is based on two papers by Buchwald [3] and Buchwald and Tran Cong [4] who studied the diffraction of elastic waves by a small circular cavity and a small elliptic cavity, respectively, in an otherwise unbounded domain. Here we clarify and systematize some aspects of their work and extend it to the diffraction of elastic waves by a small cylindrical cavity with a hypotrochoidal boundary. Results for the case of an incident P-wave are compared, in the special case of an elliptic boundary, with the results from the numerical solution of the boundary integral equation method.

More and more experimental evidence demonstrates that the slip boundary condition plays an important role in the study of nano- or micro-scale fluid. We propose a homogenization approach to study the effective slippage problem. We show that the effective slip length obtained by homogenization agrees with the results obtained by the traditional method in the literature for the simplest Stokes flow; then we use our approach to deal with two examples which seem quite hard by other analytical methods. We also include some numerical results to validate our analytical results.

We discuss the separability of the Hamilton-Jacobi equation for the Kerr metric. We use a recent theorem which says that a completely integrable geodesic equation has a fully separable Hamilton-Jacobi equation if and only if the Lagrangian is a composite of the involutive first integrals. We also discuss the physical significance of Carter's fourth constant in terms of the symplectic reduction of the Schwarzschild metric via SO(3), showing that the Killing tensor quantity is the remnant of the square of angular momentum.