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.

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 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.

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.

We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T: X → X on some Banach space X with the following property: T has a bounded H∞-functional calculus with respect to the unit disc 𝔻(that is, T is polynomially bounded) but T does not have any bounded H∞-functional calculus with respect to a Stolz domain of 𝔻 with vertex at 1. Also we show that for an R-Ritt operator the unconditional Ritt condition of Kalton and Portal is equivalent to the existence of a bounded H∞-functional calculus with respect to such a Stolz domain.

We provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

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.

The existence of solutions to equations in normed spaces is proved when the nonlinear part of the equation satisfies growth and asymptotic conditions, whether the linear part is invertible or not. For this, we use the coincidence degree theory developed by Mawhin. We apply our abstract results to boundary value problems for nonlinear vector ordinary differential equations. In particular, we consider the Picard boundary value problem at the first eigenvalue and the periodic boundary value problem at resonance. In both cases, the nonlinear term can be of superlinear type. Also, necessary and sufficient conditions of Landesman-Lazer type are obtained.

In this study, we are concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes Korteweg equations of a compressible fluid in the half space. We assume that the space-asymptotic states and the boundary data satisfy some conditions so that the time-asymptotic state of this solution is a rarefaction wave. Then we show that the rarefaction wave is non-linearly stable, as time goes to infinity, provided that the strength of the wave is weak and the initial perturbation is small. The proof is mainly based on $L^{2}$-energy method and some time-decay estimates in $L^{p}$-norm for the smoothed rarefaction wave.

Radial deformations of a ball composed of a nonlinear elastic material and corresponding to cavitation have been much studied. In this paper we use rescalings to show that each such deformation can be used to construct infinitely many non-symmetric singular weak solutions of the equations of nonlinear elasticity for the same displacement boundary-value problem. Surprisingly, this property appears to have been unnoticed in the literature to date.

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.

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.

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.

The aim of this paper is to study a boundary time-optimal control problem for the heat equation in a two-dimensional ball. The main ingredient is the extension of a result concerning Müntz polynomials due to Borwein and Erdélyi that allows us to prove an observability inequality for the dynamical system's truncation to a finite number of modes. This result, combined with a well-known Lebeau–Robbiano argument used to show the null-controllability of parabolic type equations, enables us to deduce the existence, uniqueness and bang-bang properties for the boundary time-optimal control.

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

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 presents a systematic account of ℤ/2-equivariant KO-theoretic methods in the study of r-fields (that is, r linearly independent cross-sections, r ≧ 1) on a real vector bundle. Applications of the theory to the problem of immersing complex and quaternionic projective spaces in Euclidean space are given in [13].

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.

We present a new method for solving the minimization problem in ferromagnetism. Our method is based on replacing the non-local non-convex total energy of magnetization by a new local non-convex energy of divergence-free fields. Such a general method works in all dimensions. However, for the two-dimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some recent non-convex minimization procedures in the calculus of variations. We focus on the two-dimensional case in this paper and leave the three-dimensional situation to future work. Special emphasis is placed on the analysis of the existence/non-existence depending on the applied field and the physical domain.