Forms of the colour algebra introduced by Domokos and Kövesi-Domokos are studied by relating them to the well-known Cayley–Dickson algebras. Automorphisms groups and derivation algebras of these algebras are also determined.

The Rolandic cortex is discussed in the following paper, and an account is given of experiments on monkeys, performed with the object of mapping out this region more accurately and of delimiting in it the areas concerned with movements of different parts of the body. The method employed is a new one, and will be described in detail.

This paper is a continuation of two papers on Seismic Radiations published in the Proceedings of the Royal Society of Edinburgh, vol. xxviii, pp. 217–230 (1907–8) and vol. xxx, pp. 23–37 (1909). The object of the present communication is to place on record a new determination of the laws of propagation of seismic waves based upon a method of calculation in which no assumptions are made as to the functional relation between velocity of propagation and distance from the earth's centre. References to the work of others will be given incidentally as occasion arises.

Pullback attractors with forwards unbounded behaviour are to be found in the literature, but not much is known about pullback attractors with each and every section being unbounded. In this paper, we introduce the concept of unbounded pullback attractor, for which the sections are not required to be compact. These objects are addressed in this paper in the context of a class of non-autonomous semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as the initial time goes to -∞, for each elapsed time. Distinct regimes for the non-autonomous term are taken into account. Namely, we address the small non-autonomous perturbation and the asymptotically autonomous cases.

Using the theory of generalised characteristics, we study the structure of BV solutions of genuinely nonlinear systems of two conservation laws whose shock and rarefaction wave curves of the first family are straight lines. We also establish a priori estimates on the variation of the solution similar to those obtained earlier by Glimm and Lax.

Decay estimates for the energy are derived for the initial boundary value problem of the wave equation with a degenerate dissipative term:

where Ω is a bounded domain in Rn, a(×) is a nonnegative function such that a1 ∊Lp(Ω) for some p > 0 and f is a function tending to 0 rapidly as t → ∞

Formulae are derived for the first two approximations to the radiation pattern of a penetrable body in the long wavelength limit. They are expressed in terms of a solution of Laplace's equation satisfying certain boundary conditions. Various inequalities and principles which help fix this solution are given.

A monotone iteration scheme for the solution of the initial boundary problems associated with a system of semilinear parabolic differential equations has been developed that does not require the nonlinearities to be quasimonotone. The class of equations to which this scheme applies includes physical models that describe combustion processes involving Arrhenius reaction terms.

This paper combines the global bifurcation theory of Rabinowitz with Sturmian theory and careful estimates to obtain a detailed qualitative description of bifurcating branches of solutions to the equations for whirling nonuniform, nonlinearly elastic strings. These results generalize earlier work of Kolodner and Stuart on inextensible strings. It is shown that the location of solution branches for the generalization of Kolodner's problem is especially sensitive to the material properties of the string, whereas that for Stuart's problem is not. The analysis of a third problem illuminates the source of this dichotomy.

We consider the following problem:

\begin{alignat*}{2} -\text{div}(A(x,u)\nabla u)&=u^s+f(x) & \quad &\text{in }\varOmega, \\ u(x)&\ge0 & & \text{in }\varOmega, \\ u(x)&=0 & & \text{on }\p\varOmega, \end{alignat*}

where $\varOmega$ is an open bounded subset of $\mathbb{R}^N$, $N\ge3$, and

$$ A:\varOmega\times\mathbb{R}\rightarrow M_{N\times N} $$

is an elliptic matrix such that when $u\to\infty$ is non-coercive.

We prove that there exists a complete system of eigenvectors of the eigenvalue problem

for self-adjoint operators Tr and Vrs on separable Hilbert spaces Hr. It is assumed that

(i) the operators Tr have discrete spectrum;

(ii) the operators Vrs are bounded and commute for each r;

(iii) the operators Vrs have the definite sign factor property.

This theorem generalizes and improves a result of Cordes for two-parameter problems. The proof of the theorem depends on an approximation of the given eigenvalue problem by simpler problems, a technique which is related to Atkinson's proof of his expansion theorem.

The principle of similar structures was first enunciated by Galileo. Applying it mechanically, it appears that the strength of a structure varies as the square of the linear dimensions, and the weight as the cube. This principle and its mechanical applications have been widely illustrated by reference to the bodies of animals, and many of their peculiarities are necessary consequences of its effect in the course of their evolution. For instance, the columnar legs of the elephant or of the moa are held to be the inevitable sequel to the large size and consequent weight of those animals, while the thin arched legs of insects are only possible where the body itself is small and light.

The principle followed in the reductions which form the subject of this communication may be briefly stated thus:—

The varying temperature during a year, shown by any one of the underground thermometers on an average for a series of years, is expressed by the ordinary method in a trigonometrical series of terms representing simple harmonic variations,—the first having a year for its period, the second a half year, the third a third part of a year, and so on. The yearly term of the series is dealt with separately for the thermometers at the different depths, the half yearly term also separately, and so on, each term being treated as if the simple periodic variation which it represents were the sole variation experienced.

We consider periodic homogenization problems for Lévy operators with asymmetric Lévy densities. The formal asymptotic expansion used for the α-stable (symmetric) Lévy operators (α ∈ (0, 2)) is not directly applicable to such asymmetric cases. We rescale the asymmetric densities and extract the most singular parts of the measures, which average out the microscopic dependencies in the homogenization procedures. We give two conditions, (A) and (B), that characterize such a class of asymmetric densities under which the above ‘rescaled’ homogenization is available.

After stating the case distinctly, and dwelling emphatically on its importance, as shewn by individual instances in private life, and by the statistics of the world at large, the author proceeded to describe the various methods adopted at present in India, and shewed their incapacity to meet the end proposed, as they merely agitated the air already in a room, or perniciously overloaded it with moisture.

It is shown that, for integrals of the type with Ω RN open, bounded and f: Ω × Rm × Rd → [0, + ∞) Carathéodory satisfying a growth condition 0 ≤ f(x, u, υ) ≤ C(1 + |υ|p), for some p ∈ (1, + ∞), a sufficient condition for lower semi-continuity along sequences un → u in measure, υn → υ in Lp, Aυn → 0 in W−1, p is the Ax-quasi-convexity of f(x, u, ·). Here, A is a variable coefficients operator of the form with A(i) ∈ C∞ (Ω; Ml × d) ∩ W1, ∞, i = 1, …, N, satisfying the condition and Ax denotes the constant coefficients operator one obtains by freezing x. Under additional regularity conditions on f, it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences {υn} in Lp satisfying the condition Aυn → 0 in W−1,p, is obtained.

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions have to be imposed only in an increasing sequence of annular regions surrounding the origin.

In the context of reliability theory, two definitions are given for coherent functions of n variables, where both function and variables can take any of l possible levels. The enumeration problem for such functions is discussed and several recursive bounds are derived. In the case of l = 2 (the Dedekind problem) a recursive upper bound is derived which is better than the previous best explicit upper bound forn < 15, and also provides a systematic improvement on this bound for larger values of n.

We study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation

We establish some eigenvalue criteria for the existence of non-trivial T-periodic solutions of a class of first-order functional differential equations with a nonlinearity f(x). The nonlinear term f(x) can take negative values and may be unbounded from below. Conditions are determined by the relationship between the behaviour of the quotient f(x)/x for x near 0 and ±∞ and the smallest positive characteristic value of an associated linear integral operator. This linear operator plays a key role in the proofs of the results and its construction is non-trivial. Applications to related eigenvalue problems are also discussed. The analysis mainly relies on the topological degree theory.