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.

A stochastic model predictive control (SMPC) algorithm is developed to solve the problem of three-dimensional spacecraft rendezvous and docking with unbounded disturbance. In particular, we only assume that the mean and variance information of the disturbance is available. In other words, the probability density function of the disturbance distribution is not fully known. Obstacle avoidance is considered during the rendezvous phase. Line-of-sight cone, attitude control bandwidth, and thrust direction constraints are considered during the docking phase. A distributionally robust optimization based algorithm is then proposed by reformulating the SMPC problem into a convex optimization problem. Numerical examples show that the proposed method improves the existing model predictive control based strategy and the robust model predictive control based strategy in the presence of disturbance.

A calculation of the electromagnetic response of a thin conductor in the presence of an exciting primary magnetic field has been attempted by various authors. Analytic solutions are obtainable when either the conductor is of infinite extent or when the problem possesses some symmetry. The loss of symmetry makes the problem difficult to solve except for the simplest shape – that of a circular conductor. A numerical method has been used for the rectangular conductor by other authors. In this paper we consider the response due to a thin plane conductor of arbitrary shape. The method involves the numerical generation of a set of body-fitted orthogonal curvilinear coordinates which maps the conductor onto a unit square. Good orthogonal grids can be generated for shapes that do not deviate too far from the rectangular. In terms of these curvilinear coordinates the vector potential for the area current density satisfies an integro-differential equation which is solved numerically.

Connections between a linear partial difference equation with constant coefficients and a nonlinear partial difference equation are established by means of a comparison theorem and a continuous dependence of parameters theorem. A linearized oscillation theorem is also established as an application.

We present a general closed 4-point quadrature rule based on Euler-type identities. We use this rule to prove a generalization of Hadamard's inequalities for (2r)-convex functions (r ≥ 1).

It is shown how to decompose a three-dimensional field periodic in two Cartesian coordinates into five parts, three of which are identically divergence-free and the other two orthogonal to all divergence-free fields. The three divergence-free parts coincide with the mean, poloidal and toroidal fields of Schmitt and Wahl; the present work, therefore, extends their decomposition from divergence-free fields to fields of arbitrary divergence. For the representation of known and unknown fields, each of the five subspaces is characterised by both a projection and a scalar representation. Use of Fourier components and wave coordinates reduces poloidal fields to the sum of two-dimensional poloidal fields, and toroidal fields to the sum of unidirectional toroidal fields.

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.

This paper presents a new immersed finite volume element method for solving second-order elliptic problems with discontinuous diffusion coefficient on a Cartesian mesh. The new method possesses the local conservation property of classic finite volume element method, and it can overcome the oscillating behaviour of the classic immersed finite volume element method. The idea of this method is to reconstruct the control volume according to the interface, which makes it easy to implement. Optimal error estimates can be derived with respect to an energy norm under piecewise $H^{2}$ regularity. Numerical results show that the new method significantly outperforms the classic immersed finite volume element method, and has second-order convergence in $L^{\infty }$ norm.

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 extends earlier work by using the factorisation method to discuss solutions of period four for the difference equation

This equation was suggested by R. M. May as a simple mathematical model for the effect of frequency-dependent selection in genetics. It is shown that for a given value of the parameter, a, the identification of solutions of period four can be reduced to finding real roots for a polynomial equation of degree eight. The appropriate values of xn follow from a quartic equation. By splitting up the problem in this way it becomes relatively straightforward to determine the critical values of a at which the various solutions of period four first appear and to discuss the stability of these solutions. Intervals of stability are tabulated in the paper.

A study is made of a non-linear diffusion equation which admits bifurcating solutions in the case where the bifurcation is asymmetric. An analysis of the initial-value problem is made using the method of multiple scales, and the bifurcation and stability characteristics are determined. It is shown that a suitable interpretation of the results can lead to determination of the choice of the bifurcating solution adopted by the system.

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.

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.

A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components.

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.

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.

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.

The quasi-linear infiltration problem of flow from a semi-infinite wetted region on a soil of finite depth above a horizontal water table is considered in the presence of linearised evaporative loss away from the region. The resulting equations are solved by the Wiener-Hopf technique in terms of certain infinite products. Expressions for the porosity and stream function are derived, and appropriately plotted throughout the layer.

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.