We study a periodic Kolmogorov model with m predators and n prey. By means of the comparison theorem and a Liapunov function, a set of easily verifiable sufficient conditions that guarantee the existence, uniqueness and global attractivity of the positive periodic solution is obtained. Finally, some suitable applications are given to illustrate that the conditions of the main theorem are feasible.

The probability that an interval I is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval I includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxiliary quantities. We summarize and extend earlier work by expressing the probability and some of the auxiliary quantities in terms of Painlevé transcendents.

In this work we shall study the existence of extremal solutions for an impulsive problem with functional-boundary conditions and weak regularity assumptions, not only on the right-hand side of the equation and on the functions that define the boundary conditions, but also on the impulse functions, which will be required to be nondecreasing, but not continuous as well, as is customary in the literature.

Moreover, in order to prove one of our results we shall study a general impulsive linearproblem, giving a complete characterisation of resonance for it.

We consider the dynamical characteristics of a continuous-time isolated Hopfield-type neuron subjected to an almost periodic external stimulus. The model neuron is assumed to be dissipative having finite time delays in the process of encoding the external input stimulus and recalling the encoded pattern associated with the external stimulus. By using non-autonomous Halanay-type inequalities we obtain sufficient conditions for the hetero-associative stable encoding of temporally non-uniform stimuli. A brief study of a discrete-time model derived from the continuous-time system is given. It is shown that the discrete-time model preserves the stability conditions of the continuous-time system.

We are concerned with the solvability of variational inequalities that contain degenerate elliptic operators. By using a recession approach, we find conditions on the boundary conditions such that the inequality has at least one solution. Existence results of Landesman-Lazer type for a nonsmooth inequality and a resonance problem for a weighted p-Laplacian are discussed in detail.

We consider a single server queueing system where each customer visits the queue a fixed number of times before departure. A customer on his j th visit to the queue is defined to be a class-j -customer. We obtain the joint probability generating function for the number of class-j-customers and also obtain the Laplace-Stieltjes transform for the total response time of a customer.

The Maxwell-Dirac equations model an electron in an electromagnetic field. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial differential equations (PDE's). Well-behaved solutions, within reasonable Sobolev spaces, have been shown to exist globally as recently as 1997 [12]. Exact solutions have not been found—except in some simple cases.

We have shown analytically in [6, 18] that any spherical solution surrounds a Coulomb field and any cylindrical solution surrounds a central charged wire; and in [3] and [19] that in any stationary case, the surrounding electron field must be equal and opposite to the central (external) field. Here we extend the numerical solutions in [6] to a family of orbits all of which are well-behaved numerical solutions satisfying the analytic results in [6] and [11]. These solutions die off exponentially with increasing distance from the central axis of symmetry. The results in [18] can be extended in the same way. A third case is included, with dependence on z only yielding a related fourth-order ordinary differential equation (ODE) [3].

A variation-of-constants formula is obtained for a linear abstract evolution equation in Hilbert space with unbounded perturbation and free term. As an application, the state of the abstract controlled system with unbounded mixed controls is explicitly given.

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.

Daniel et al. [6] analysed the singularity structure of the continuum limit of the one-dimensional anisotropic Heisenberg spin chain in a transverse field and determined the conditions under which the system is nonintegrable and exhibits chaos. We investigate the governing differential equations for symmetries and find the associated first integrals. Our results complement the results of Daniel et al.

In this work the modified Green's function technique for an exterior Dirichlet and Neumann problem in linear elasticity is investigated. We introduce a modification of the fundamental solution in order to remove the lack of uniqueness for the solution of the boundary integral equations describing the problems, and to simultaneously minimise their condition number. In view of this procedure the cases of the sphere and perturbations of the sphere are examined. Numerical results that demonstrate the effect of increasing the number of coefficients in the modification on the optimal condition number are also presented.

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 propose a new trust region algorithm for solving the system of nonsmooth equations F(x) = 0 by using a smooth function satisfying the Jacobian consistency property to approximate the nonsmooth function F(x). Compared with existing trust region methods for systems of nonsmooth equations, the proposed algorithm possesses some nice convergence properties. Global convergence is established and, in particular, locally superlinear or quadratical convergence is obtained if F is semismooth or strongly semismooth at the solution.

The classical dynamics of non-relativistic particles are described by the Schrödinger wave equation, perturbed by quantum potential nonlinearity. Quantization of this dispersionless equation, implemented by deformation of the potential strength, recovers the standard Schrödinger equation. In addition, the classically forbidden region corresponds to the Planck constant analytically continued to pure imaginary values. We apply the same procedure to the NLS and DNLS equations, constructing first the corresponding dispersionless limits and then adding quantum deformations. All these deformations admit the Lax representation as well as the Hirota bilinear form. In the classically forbidden region we find soliton resonances and black hole phenomena. For deformed DNLS the chiral solitons with single event horizon and resonance dynamics are constructed.

In this paper, we consider the problem of the steady-state fully developed magnetohydrodynamic (MHD) flow of a conducting fluid through a channel with arbitrary wall conductivity in the presence of a transverse external magnetic field with various inclined angles. The coupled governing equations for both axial velocity and induced magnetic field are firstly transformed into decoupled Poisson-type equations with coupled boundary conditions. Then the dual reciprocity boundary element method (DRBEM) [20] is used to solve the Poisson-type equations. As testing examples, flows in channels of three different crosssections, rectangular, circular and triangular, are calculated. It is shown that solutions obtained by the DRBEM with constant elements are accurate for Hartmann number up to 8 and for large conductivity parameters comparing to exact solutions and solutions by the finite element method (FEM).

We investigate the reflection and transmission of SH-waves at a corrugated interface between two different anisotropic, heterogeneous elastic solid half-spaces. Both the media are assumed to be transversely isotropic and vertically heterogeneous. Rayleigh's method is followed and expressions for the reflection and transmission coefficients are obtained in closed form for the first-order approximation of the corrugation. It is found that these coefficients depend on corrugation and are affected by the anisotropy and heterogeneity of the media. Numerical computations for a particular model have been performed.

We give an intrinsic construction of a coupled nonlinear system consisting of two first-order partial differential equations in two dependent and two independent variables which is determined by a hyperbolic structure on the complex special linear group regarded as a real Lie group G. Despite the fact that the system is not Darboux semi-integrable at first order, the construction of a family of solutions depending.upon two arbitrary functions, each of one variable, is reduced to a system of ordinary differential equations on the 1-jets. The ordinary differential equations in question are of Lie type and associated with G.

Elastic behaviour of a nonhomogeneous transversely isotropic half-space is studied under the action of a smooth rigid axisymmetric indentor. Hankel transforms of different orders have been used. It is observed that in contrast to a homogeneous medium, the pressure distribution in the contact region in a nonhomogeneous medium is not directly available, rather it is obtainable from the solution of a Fredholm integral equation. The integral equation is solved for a flat-ended punch and paraboloidal indentations for various values of the nonhomogeneity parameter, and the effects of nonhomogeneity in elastic behaviour on stresses have been shown graphically. The results of the associated homogeneous case are readily available from the results of the present study.

We present various inequalities for Euler's beta function of n variables. One of our theorems states that the inequalities

hold for all xi ≥ (i = 1,… n; n ≥ 3) with the best possible constants an = 0 and bn = 1 − 1/(n − 1)!. This extends a recently published result of Dragomir et al., who investigated (*) for the special case n = 2.