This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.

It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

The paper considers the small time evolution of the interface which appears in the boundary value problem

A diffusion dominated small time outer expansion is not uniformly valid when x = O(t½ log t) and has to be supplemented by an inner expansion valid in a region where diffusion and absorption are of equal importance. For p < 1 a zero order inner solution is constructed, which is consistent with a moving interface where u is zero, and the small time development of this is given.

These results are first derived in a heuristic way using the method of matched expansions. However, an important aim of the paper is to give rigorous proofs, being one of the very few cases in nonlinear diffusion where this has been done.

The entire collection comprises from 150 to 200 species, the majority of which are new to science.

As yet only some of the more abnormal Synascidiœ and about half the Ascidiœ simplices have been carefully examined. The present paper is the preliminary report on the ASCIDIÆ, the first family of the Ascidiœ simplices.

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.

In many problems in physics and chemistry certain determinants of large order and of a special type require to be evaluated. The determinants in question usually arise from some kind of secular equation, and in most cases they owe their origin to a system of particles of one kind or another which, in their equilibrium positions, form a regular lattice, each particle being acted upon by its nearest neighbours and perhaps by a fixed boundary. These forces may be assumed to be elastic in character, to the degree of approximation required. For simplicity we shall refer to these particles as atoms although they may be electrons, molecules or even material particles in the Newtonian sense, according to the nature of the problem under investigation. In particular we may mention the occurrence of these determinants in problems involving the solution of Schrödinger's Wave Equation for permitted energy levels and in determining the distribution of electric charge in crystals, metals and large molecules. They have also been used by Born in his investigations of crystal structure by means of X-rays. It is hoped therefore that the results achieved in this paper will be of interest to the physicist and chemist as well as to the mathematician.

In the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

Sharp weak type (1,1) and Lp estimates in dimension one are obtained for uncentred maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions, uncentred maximal functions fail to satisfy such estimates. Analogous results for centred maximal functions are given in all dimensions.

With certain initial and boundary conditions the solution u* to the semilinear heat equation ∂u*/∂t = ∂u* + λ * f(u*), where f is a positive superlinear function and λ is the supremum of the open spectrum for the steady state problem Δw + λf(w) = 0, is found to exist for all time and to be unbounded. Moreover u* approaches w* a singular steady state, as / tends to infinity.

We consider the existence of positive solutions for the boundary-value problem

(q(t)ϕ(u′))′ + λf(t,u) = 0, r < t < R,

au(r) − bϕ−1 (q(r))u′(r) = 0, cu(R) + dϕ−1(q(R))u′(R) = 0,

where ϕ(u′) = |u′|p−2u′, p > 1, λ > 0, f is p-superlinear or p-sublinear at ∞ and is allowed to become −∞ at u = 0. Our results unify and extend many known results in the literature.

In 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.

The results obtained may be summarised in the following table comparing the filial and fraternal coefficients for autosomal and sex-linked variation. When dominance is complete the values given are referred to equal frequency of the allelomorphic genes, i.e.p = ½ = q.

Briefly, then, the contribution of sex-linked genes to the observed variance will tend to lower the correlation between father and son and between brother and sister. It will raise the correlation between father and daughter or between sister and sister conspicuously. These considerations emphasise the interest attaching to a separate investigation of fraternal correlation for sibs of all three sex classes in connection with studies on twins. If the correlation for dizygotic twins of like sex is higher than for dizygotic twins of unlike sex, the difference might be due to the greater similarity of environment for two individuals of the same sex, or it might be due to the contribution of sex-linked genes to the observed variance. In the latter case the correlation for boy pairs should be significantly lower than the correlation for girl pairs. The wide discrepancies in recorded data for fraternal correlations with respect to intelligence quotients might be re-examined from this standpoint. If an appreciable proportion of the total variance is due to sex-linked genes, it is of more importance that a boy should have a clever mother than a clever father.

The hyperbolic system of conservation laws that govern the motion of a homogeneous isotropic, nonlinearly elastic body is shown to have a discontinuous solution for a class of stored-energy functions of slow growth. This solution is admissible by the usual entropy criterion and is in fact preferred by the entropy-rate criterion over the smooth equilibrium solution to the same problem. The existence of such a dissipative solution shows that the equilibrium solution is dynamically unstable. This instability cannot be ascertained by linearisation.

We have, during the last two years, made a series of observations upon specimens of Sagartia troglodytes which are at least fifty years old, and have thought it worth while to give a somewhat detailed account of these, as, so far as we can ascertain, there is only one other recorded case of longevity in Coelenterates (see p. 302), and very few in the whole of the Invertebrata.

The aim of this paper is to study Hamiltonian elliptic system of the form 0.1

$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$

0.2

$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$

I much regret it has not been possible for me, during this summer, experimentally to determine the best forms and sizes of the details in the construction of the C screen; nor have I been able to keep a continuous record of its readings. Only a few observations have been made at intervals with it, as originally constructed, and shown in fig. 1 of this paper, and with the Stevenson screens. The result of all these trials is to confirm the conclusions already arrived at. The C screen always gave the lowest readings when there was any radiation.

We study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

Two theorems on the spectrum of operator pencils generated by the three-dimensional system of anisotropic elasticity in the complement of a plane angle are proved.

Let Ω ⊂ RN be a bounded domain. We consider the nonlinear problem and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative solutions of the following linear problem: .

In particular, if p > (N + 2)/(N − 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in § 1 of this paper).

Vector-borne diseases, such as chikungunya, dengue, malaria, West Nile virus, yellow fever and Zika, pose a major global public health problem worldwide. In this paper we investigate the propagation dynamics of diffusive vector-borne disease models in the whole space, which characterize the spatial expansion of the infected hosts and infected vectors. Due to the lack of monotonicity, the comparison principle cannot be applied directly to this system. We determine the spreading speed and minimal wave speed when the basic reproduction number of the corresponding kinetic system is larger than one. The spreading speed is mainly estimated by the uniform persistence argument and generalized principal eigenvalue. We also show that solutions converge locally uniformly to the positive equilibrium by employing two auxiliary monotone systems. Moreover, it is proven that the spreading speed is the minimal wave speed of travelling wave solutions. In particular, the uniqueness and monotonicity of travelling waves are obtained. When the basic reproduction number of the corresponding kinetic system is not larger than one, it is shown that solutions approach to the disease-free equilibrium uniformly and there is no travelling wave solutions. Finally, numerical simulations are presented to illustrate the analytical results.