A 3 × 3 autonomous, non-linear system of ordinary differential equations modelling the immune response in animals to invasion by active self-replicating antigens has been introduced by G. I. Bell and studied by G. H. Pimbley Jr. Using Hopf's theorem on bifurcating periodic solutions and a stability criterion of Hsu and Kazarinoff, we obtain existence of a family of unstable periodic solutions bifurcating from one steady state of a reduced 2×2 form of the 3×3 system. We show that no periodic solutions bifurcate from the other steady state. We also prove existence and exhibit a stability criterion for families of periodic solutions of the full 3×3 system. We provide two numerical examples. The second shows existence of orbitally stable families of periodic solutions of the 3×3 system.

We consider the stationary equations of a general viscous fluid in an infinite (periodic) slab supplemented with Navier's boundary condition with a friction term on the upper part of the boundary. In addition, we assume that the upper part of the boundary is described by a graph of a function φε, where φε oscillates in a specific direction with amplitude proportional to ε. We identify the limit problem when ε → 0, in particular, the effective boundary conditions.

There is a large catalogue of decompositions of conditioned superprocesses in terms of an ‘immortal backbone’ or ‘skeleton’ along the branches of which mass is constantly immigrated. We add to this with a study of the (infinite-variance) (1 + β)-superprocess, conditioned on survival until some fixed time T. As one would expect, we see a Poisson number of immortal trees (conditioned on there being at least one), along which mass (conditioned to die before time T) is immigrated. However, here we see a new source of immigration. Not only is mass immigrated along the branches of the immortal trees, but also there is an extra burst of immigration whenever the immortal tree branches. Moreover, the rate of immigration along the branches is no longer deterministic. In the limit as T → ∞, the immortal trees degenerate to the Evans immortal particle and the immigration (of unconditioned mass) along the particle is dictated by a stable subordinator.

It is well known that quaternions are intimately connected with spherical trigonometry, and in fact they reduce that subject to a branch of algebra. The question is suggested whether there is not a system of quaternions complementary to that of Hamilton, which is capable of expressing trigonometry on the surface of the equilateral hyperboloids. The rules of vector-analysts are approximately complementary to those of quaternions. In this paper I propose to show how they can be made completely complementary, and that, when so rectified, they yield the hyperbolic counterpart of the spherical quaternions.

As in an earlier paper by the author, three cardinal numbers, the shift, the defect and the collapse, are associated with each element of the full transformation semigroup ℑ(X), where X is an infinite set. It is shown that the elements of finite shift and non-zero defect form a subsemigroup F of ℑ(X). Moreover, if E(F) denotes the set of idempotents in F then 〈E(F)〉 = F, but (E(F))n ⊂F for every finite n. For each infinite cardinal m not exceeding ∣X∣ the set Qm of balanced elements of weight m, i.e. those with shift, defect and collapse all equal to m, forms a subsemigroup of ℑ(X). Moreover, (E(Qm))4=Qm,(E(Qm))3⊂Qm.

Let q be a positive integer and let (an) and (bn) be two given ℂ-valued and q-periodic sequences. First we prove that the linear recurrence in ℂ 0.1

$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$

0.2

$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$

0.3

$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$

0.4

$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$

0.5

$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are non-degenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

The characteristic numbers of the Mathieu equation

are those values of a for which, when q is given, the equation admits of a solution of period π or 2π. The periodic solutions, or Mathieu functions, may be developed as a Fourier-series convergent for all values of q,

multiplied by one or other of the factors

We study the existence and asymptotic behaviour of positive solutions of a semilinear elliptic equation in entire space. A special case of this equation is the scalar curvature equation which arises in Riemannian geometry.

Integral transformations T of the formal form

where G is a Meijer-G function are studied on the spaces ℐμ,r of functions f such that

Results are obtained on the boundedness, representation and range of the transformations, and on when they have an inverse of the same formal form.

The paper is concerned with the distributional properties of Markoff chains in two and three dimensions where the transition probability for the length of a step and its orientation relative to that of the previous step is specified.

The discrete two-dimensional chain of n steps is first discussed, and by the use of moving axes an equation relating characteristic functions of the end-point distribution for successive values of n is obtained. The corresponding differential equation for the limiting chain with continuous first derivatives is given and asymptotic solutions for long chains are found.

The three-dimensional chain is similarly treated in terms of moving axes, and the limiting continuous chain is again discussed. Finally the same methods are applied to the discrete chain of equal steps to obtain the asymptotic form of the end-point distribution for long chains.

Regular Sturm–Liouville problems with indefinite weight functions may possess finitely many non-real eigenvalues. In this paper we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression.

We deal here with a mixed (hyperbolic-elliptic) system of two conservation laws modelling phase-transition dynamics in solids undergoing phase transformations. These equations include nonlinear viscosity and capillarity terms. We establish general results concerning the existence, uniqueness and asymptotic properties of the corresponding travelling wave solutions. In particular, we determine their behaviour in the limits of dominant diffusion, dominant dispersion or asymptotically small or large shock strength. As the viscosity and capillarity parameters tend to zero, the travelling waves converge to propagating discontinuities, which are either classical shock waves or supersonic phase boundaries satisfying the Lax and Liu entropy criteria, or else are undercompressive subsonic phase boundaries. The latter are uniquely characterized by the so-called kinetic function, whose properties are investigated in detail here.

We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. More precisely, we study a linear variational problem related to the Poincaré inequality and to the Hardy inequality for maps in H01(Ω), where Ω is a bounded domain in ℝN, N ≥ 2, with 0 ∈ ∂Ω. In particular, we give sufficient and necessary conditions so that the best constant is achieved.

We study the monotonicity for certain combinations of generalized elliptic integrals, thus generalizing analogous well-known results for classical complete elliptic integrals, and prove a conjecture put forward by Heikkala, Vamanamurthy and Vuorinen.

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problem

where Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equation

for some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

In theoretical analyses of the hereditary composition of human populations it is often assumed that crossing occurs at random and that individuals with different hereditary make‐up are equally fertile. This state of things has been called amphimixis. Sometimes such assumptions tally so closely with the actual state of things that it is possible to employ calculations carried out from this point of departure. In other cases processes present in populations cause larger or smaller deviations from amphimixis.

In papers which the authors have already communicated to this Society, they have shown that there exists a very marked difference between the action of bases containing nitrogen as a triad and those in which that element is pentad. This result was obtained from an examination of the physiological action of substances having an unknown and complex constitution. It appeared to be necessary to institute a series of experiments with bodies having a simpler and fully known constitution. For this purpose, the salts of ammonia, tri-methylamine, and tetra-methyl-ammonium were selected. A comparison of the structural formulae of the hydrochlorates of these bases shows that tri-methylamine stands in the same relation to the salts of tetramethyl-ammonium as strychnia to the salts of methyl-strychnium.

We investigate the large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas. A conclusive answer to the problem of asymptotic behaviour is given in Theorem 1.2.

We study the behaviour of non-convex functionals singularly perturbed by a possibly oscillating inhomogeneous gradient term, in the spirit of the gradient theory of phase transitions. We show that a limit problem giving a sharp interface, as the perturbation vanishes, always exists, but may be inhomogeneous or anisotropic. We specialize this study when the perturbation oscillates periodically, highlighting three types of regimes, depending on the frequency of the oscillations. In the two extreme cases, a separation of scales effect is described.