If we look for a short time at any object, and afterwards turn the eye in another direction, we see a spectral image of the form of the object first looked at.

Again, if after we have looked at any coloured object we turn the eye in another direction, we see a spectral image of a colour complimentary to that first looked at.

In addition to these spectral forms and colours, I find there is another and distinct kind of ocular spectrum, which we may call a motion spectrum. It is seen when we look first at a body in motion and afterwards direct the eye towards an object at rest. The object at rest, when seen under these conditions, seems to be in motion, and the direction of its apparent motion is the opposite of that of the moving body first looked at.

§ 1. In the case of a function of two variables f(x, y) there is, in general, no connection between the results of proceeding to the various limits, upper, lower, and intermediate, first with respect to x and then with respect to y, and first with respect to y and then with respect to x. We cannot even assert that the repeated upper limit obtained in one way is not less than the repeated lower limit obtained in the other way.

We consider initial boundary value problems for the equations of linear thermoelasticity in both bounded and unbounded domains and for both nonhomogeneous and anisotropic media. For bounded domains, it is shown that the unique solution of the problem is time-asymptotically equal to the solution of a particular initial boundary value problem which is obtained from a natural decomposition of the original initial data and which represents a (in general non-vanishing) time harmonic part. For the unbounded case similar results are obtained, but now in the sense of weak convergence which lead to the result of local energy decay: the solution tends to zero in every compactum.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition:where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

The objective is to derive a variation of constants formula for systems of functional differential equations (or delay differential equations) coupled with functional equations (or difference equations). The difficulties arise because of the constraints imposed by the functional equations.

Conical shock waves are generated as sharp conical solid projectiles fly supersonically in the air. We study such conical shock waves in steady supersonic flow using an isentropic Euler system. The stability of such attached conical shock waves for non-symmetrical conical projectiles and non-uniform incoming supersonic flow is established. Meanwhile, the existence of the solution to the Euler system with such attached conical shock as a free boundary is also proved for solid projectiles close to a regular solid cone.

The paper suggests a procedure for direct construction of minimal extensions of constrained optimisation problems, particularly those containing controls in coefficients of elliptic equations. The preliminary version of the procedure has been described in [1].

This paper is concerned with the structure of M = Maps(G), the near-ring of all mappings from a group G to itself which commute with a group S* of automorphisms of G. Here S is S* together with the zero endomorphism. Necessary and sufficient conditions on the pair (G, S) are obtained for M to be (i) regular, (ii) unit regular, (iii) an equivalence near-ring. These conditions take a very simple form. In the case (iii), the two-sided M-subgroups of M are determined. The next result shows that under suitable conditions, M is a simple near-ring. A definition of transitivity is given for subnear-rings of M, and some properties of transitive near-rings are proved. Finally two examples are given to show that all the classes of near-rings considered are distinct.

A new method is developed for obtaining the asymptotic form of solutions of the fourth-order differential equation

where m, n are integers and 1 ≦ m, n ≦ 2. The method gives new, shorter proofs of the well-known results of Walker in deficiency index theory and covers the cases not considered by Walker.

Lp–Lq decay estimates of solutions to Cauchy problems of linear thermoelastic systems with second sound in three spatial variables are studied. By carefully analysis of the different effects of higher, middle and lower frequencies in phase space, the asymptotic behaviour of characteristic roots of coefficient matrices is obtained. Then, with the help of the information on characteristic roots and the interpolation theorem, a decay estimate of parabolic type for the coupled system of the potential part of displacement, temperature and heat flux is obtained. Finally, Lp–Lq decay estimates of hyperbolic type for the original thermoelastic systems with second sound are obtained.

Let Vrs, s = 1 … k, be Hermitian operators on separable Hilbert spaces Hr, r = 1 … k. For x⊗ = x1⊗ … ⊗ xk ∈ H = H1⊗…Δx⊗ by the formal determinantal expansion ⊗ det [Vrsxr Δ is then extended by linearity and continuity to H. It is shown that a positive definiteness condition on Δ need be checked only for decomposable tensors like x⊗. Applications are given to several conditions in the multiparameter eigenvalue literature.

In this paper, we find the canonical matrices of the conjugacy classes of the Sylow p-subgroup of GL(n, pt) consisting of all upper unitriangular matrices, whose cardinality is one of the two maximal possible values, that is, pt(n−1)(n−2)/2 and ptn(n−3)/2, as well as their number.

It is only within a few months that my attention has been (at first accidentally) called to this matter. For, though I owe to the kindness of Prof. Klein a copy of his and Sommerfeld's Theorie des Kreisels, I had passed over, in reading the work, the “Digression on Quaternions” which it contains. But Prof. C. N. Little, in the course of correspondence about his remarkable paper on Knots (whose passage through the press I was looking after), referred me for a numerical detail to an article by Prof. Klein on the progress of publication of Gauss' Gesammelte Werke. Shortly afterwards Prof. Joly called my attention to the same article from another point of view. These references have led me to write the present paper; whose somewhat puzzling title is explained in the first section below.

The work embraces the results of the systematic examination, for fossil micro-organisms, of slides prepared from ironstones and ferruginous limestones from various districts of Great Britain.

Abundant traces were found in the Frodingham Ironstone of a fossil fungus allied to the modern Phycomycetes. This is named Phycomycites Frodinghamii. The traces include hyphæ, sporangia, and spores. These are described, measured, and photographed, and details of the life-history, so far as ascertainable, are given.

A new proof of the Holley-Preston generalisation of the Fortuin-Kastelyn-Ginibre inequalities is given, and Batty's extension to the case of infinite products is discussed briefly. An application of the theorems in combinatorial probability theory is described.

We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C0 and by Gowers for ℓp (1 < p < ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H1, any infinite-dimensional complex L1(μ), and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.

We consider the one-dimensional, nonlocal, evolution equation derived by De Masi et al. (1995) for Ising systems with Glauber dynamics, Kac potentials and magnetic field. We prove the existence of travelling fronts, their uniqueness modulo translations among the monotone profiles and their linear stability for all the admissible values of the magnetic field for which the underlying spin system exhibits a stable and metastable phase.

In this paper we discuss the existence and multiplicity of solutions to some perturbed bifurcation problems. By using sub and supersolution techniques along with an anti-maximum principle, simple proofs of some “well known” local results of perturbed bifurcation theory are obtained. The existence of global continua of solutions is proved by using degree theory arguments.

We consider here the radial Stefan problem with Gibbs–Thomson law, which is a classical model describing growth or melting of a spherical crystal in a surrounding liquid. We shall specialize to the cases of two and three space dimensions and discuss the asymptotic behaviour of a melting crystal near its dissolution time t* > 0. We prove here that, when the interface shrinks monotonically, the asymptotics near t = t* is of the form Here, R(t) denotes the radius of the crystal, σ is a surface tension parameter and u(r, t) represents the field temperature. An important point to be noticed is that (*) exhibits no dependence on the space dimension N, in sharp contrast with results known for the case σ = 0.

The polynomials which are orthogonal with respect to

when α> – 1, M>0 are considered when α<–1 and/or M<0. The Cauchy regularization of 〈·, ·〉 provides orthogonality and generates a Pontrjagin (Krein) space spanned by the polynomials. The polynomials are eigenfunctions associated with a self-adjoint, fourth order differential operator.