We show that lattice dynamical systems naturally arise on infinite-dimensional invariant manifolds of reaction–diffusion equations with spatially periodic diffusive fluxes. The result connects wave-pinning phenomena in lattice differential equations and in reaction–diffusion equations in inhomogeneous media. The proof is based on a careful singular perturbation analysis of the linear part, where the infinite-dimensional manifold corresponds to an infinite-dimensional centre eigenspace.

Let H be a separable Hilbert space and let CL(H) be the semigroup of continuous, linear maps from H to H. Let E+ be the idempotents of CL(H). Let Ker ɑ and Im ɑ be the null-space and range, respectively, of an element ɑ of CL(H) and let St ɑ be the subspace {x∊H: xɑ = x} of H. It is shown that 〈E+〉 = I∪F∪{i}, where

and ι is the identity map. From the proof it is clear that I and F both form subsemigroups of 〈E+〉 and that the depth of I is 3. It is also shown that the depths of F and 〈E+〉 are infinite.

We consider weak solutions to

with p > 1, q ≥ max{p − 1, 1}. We exploit the Moser iteration technique to prove a Harnack comparison inequality for C1 weak solutions. As a consequence we deduce a strong comparison principle.

Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small off-diagonal perturbations. The results are based on given asymptotic expansions of the component Hamiltonians.

The function which is discussed in the greater part of this paper was introduced by Professor E. T. Whittaker in vol. xxxv (1915), pp. 181–194 of these Proceedings. A one-valued function, analytic in the finite part of the plane save possibly for poles, is taken and a table made of the values of f(x) for x = a, a + w, a — w, a + 2w, a — 2w, … The function

called the Cardinal Function, is then shown to be the simplest function satisfying the table of values so obtained, in that it is a function which, when analysed into its periodic constituents, has no rapid oscillations.

1. If all the negative terms of the determinant ∣ a1b2c3 … ∣ be changed in sign, we obtain a symmetric function, dealt with by Borchardt and Cayley, known as a Permanent and denoted by

The more important elementary properties of such functions are given in a paper published in the Proc. Roy. Soc. Edin., xi. pp. 409–418. As might be expected, relations are found to exist between them and determinants, an important instance being the theorem of § 7 of the said paper. Another theorem, not hitherto noted, deserves now to be put on record.

2. For the case of the 2nd order it is

the truth of it being self-evident.

For the case of the 3rd order it is

which is easily verified by observing that the coefficients of a1, a2, a3 in the expression on the left-hand side are respectively

and that by the previous case each of these vanishes.

The spectral theory for non-self-adjoint elliptic boundary problems involving an indefinite weight function has only been established for the case of higher-order operators under the assumption that the reciprocal of the weight function is essentially bounded. In this paper we are concerned with the spectral theory for a case where the weight function vanishes on a set of positive measure.

The inverse problem of the determination of boundary defects in a planar conductor by a finite number of electrostatic measurements on the boundary is considered. Uniqueness results and stability estimates are proved under essentially minimal regularity assumptions on the data. Finally, Lipschitz estimates for the determination of surface linear cracks are developed.

We consider a system of integrodifferential equations

where f(t, x) and F(t, s, x, y) are almost periodic in t uniformly for parameters, and we assume that the system has a bounded solution u(t). To discuss the existence of an almost periodic solution, we consider the relationship between the total stability of u(t) with respect to a certain metric ρ and the separation condition with respect to ρ. Moreover, we discuss a sufficient condition for the existence of a positive almost periodic solution of a model of the dynamics of an n-species system.

A new method is developed for identifying real-valued coefficients r(x), p(x), and q(x) for which all solutions of the fourth-order differential equation

are L2(0, ∞). The results are compared with those derived from the asymptotic theory of Devinatz, Walker, Kogan and Rofe-Beketov.

We prove existence and uniqueness results for the solution of nonlinear elliptic boundary value problems, where the linear part of the equation is given by a second-order elliptic operator not in divergence form.

We derive necessary and sufficient conditions for a unit regular monoid to have a uniquely unit regular, idempotent separating cover.

In the present paper, we consider the following Kirchhoff type problem

$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$

$V\in C(\mathbb R^3,\mathbb R)$

$\inf _{\mathbb R^N} V=0$

A widely accepted model of nerve conduction in the squid axon is the systemof four non-linear partial differential equations developed by Hodgkin and Huxley. Under space clamp and current clamp conditions these equations are reduced to a system of ordinary differential equations.

We find that under appropriate assumptions on the functions and parameters in the resulting fourth order Hodgkin-Huxley equations there occurs a bifurcation of periodic solutions from the steady state. This bifurcation takes place as the current parameter, I, passes through a critical value.

The class of M-nilpotent rings is defined as a generalisation of the class of T-nilpotent rings. Certain results for radicals of T-nilpotent rings are shown to hold also in this larger class of rings.

In this paper we are interested in the microscopic modelling of a two-dimensional two-well problem that arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyse the Hamiltonian that was introduced by Kitavtsev et al. in 2015. It turns out that this class of Hamiltonians allows for a direct control of the discrete second-order gradients and for a one-sided comparison with a two-dimensional spin system. Using this and relying on the ideas of Conti and Schweizer, which were developed for a continuous analogue of the model under consideration, we derive a (first-order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.

The set of points (α, β) ∈ℝ2 for which the problem − δu = αu+ − βu− in Ω, u = 0 on ∂Ω(u+ = max {u, o} and u− = min {−u, 0}) has nontrivial solutions is important for the study of certain nonlinear problems. It is shown that for ‘most’ bounded domains Ω in ℝn, such a set is locally the union of a finite number of curves.

As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation

$${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$

Recent studies of lattice dynamics, with respect to the existence of global compact attractors for certain discrete evolution equations, are based on the derivation of ‘tail estimates of the solution’. We show that, due to the specific nature of the discrete nonlinear Schrödinger (DNLS) system which gives rise to a simple energy equation, the method developed by J. M. Ball is applicable, and provides an alternative proof on the existence of the compactness of the attractor for the weakly damped and driven DNLS equation considered in $\mathbb{Z}^N$, $N\geq1$, lattices, without the usage of tail estimates. The approach covers various DNLS-type equations of physical significance.

Linked equations are studied on [0,1] subject to boundary conditions of the form Results are given on existence, location, asymptotics and perturbation of the eigenvalues λj and oscillation of the eigenfunctions yi.