In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

\[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \]

$D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$

$\alpha _u,\alpha _w,\mu _u,\beta$

$\Omega \subset {\mathbb {R}}^{2}$

$\beta <1$

$(u,v,w,z)$

$(1, 0, 0, 0)$

$(L^{\infty }(\Omega ))^{4}$

$t\rightarrow \infty$

Let D be an open connected subset of the open unit ball in Ṟd, d ≧ 2. We give an estimate of the harmonic measure of ∂D∩{|x| = 1} with respect to D. This estimate depends in a simple way on the geometry of D. An essential tool is a rearrangement theorem for differential inequalities. When d = 2, examples are given which illustrate the precision of the results.

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

We consider the bifurcation of positive solutions of the two-point boundary value problem

where λ> 0 is a real bifurcation parameter, and f ∊ C2 satisfies (fl) f(0) < 0, (f2) f′(s) > 0 for s > 0, (f3) f″(s) < 0 for s > 0 and (f4) limS→+∞f(s) = M where 0 < M ≦+∞. This problem has been studied by Casto and Shivaji under two additional hypotheses (f5) lims→+∞sf′(s) = 0, and (f6)f(θ)/θ < f′(θ), where θ is a positive number satisfying Assuming (fl)−(f6), Castro and Shivaji obtain some existence and nonexistence results and hence partial information on the bifurcation diagram, and they conjecture that this problem has at most two positive solutions. We prove this conjecture. Furthermore, we are able to generalise and improve their results under hypotheses (fl)−(f4). As a corollary, we show that there exists μ1 > 0 such that there exist no positive solutions for 0 <λ <μ1 and at most two positive solutions for μ1≦λ< + ∞, which improves a result of Brunovsky and Chow.

The existence of periodic solutions is studied for certain singularly perturbed differential inclusions. Applications are given to dry friction problems.

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

We construct nontrivial, non-negative quasiconvex functions denned on M2×2 with p-th order growth such that the zero sets of the functions are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. We assume that the graphs do not have rank-one connections with the Lipschitz constants sufficiently small. In particular, we are able to construct quasiconvex functions which are homogeneous of degree p (p > 1) and ‘conjugating’ invariant.

In any discussion of the zeros and turning points of the elliptic-cylinder function it is not necessary to define them more precisely than as functions of period π or 2π which satisfy the Mathieu equation

for the corresponding characteristic values of a. To simplify the following discussion, it is convenient merely to impose the condition that y(0)>0 if y(x) is even, and y′(0)>0 if y(x) is odd.

The Fulcher Bands of Hydrogen have recently been arranged by Richardson. An examination of Richardson's arrangement in the light of recent accurate measurements of the hydrogen spectrum by Gale, Monk, and Lee, while strongly confirming Richardson's allocation of the Q branches, has failed to yield confirmation of the remaining branches which he gives.

The present paper gives a number of combination relations which have been found to hold between the lines of the Fulcher Bands. These indicate that there are three main branches in each band, designated the R', Q, and P' branches, the Q branch being identical with that of Richardson. The R' and P' branches have a common initial level differing from that of the Q branch, while all three branches have a common final level. The arrangement yields term differences consistent throughout the bands.

As far as these term differences have been investigated, they have been found to be in agreement with the new mechanics.

The present investigation was suggested to me by Prof. H. Stanley Allen of St Andrews, and my warmest thanks are due to him for constant encouragement and helpful criticisms and suggestions.

In a previous communication the electrosynthesis of n-dotriacontanedi-carboxylic acid was described. This work has been continued, and three more acids of this series have been prepared, namely, eicosanedicarboxylic acid, HOOC. (CH2)20. COOH; tetracosanedicarboxylic acid, HOOC. (CH2)24. COOH; and octacosanedicarboxylic acid, HOOC. (CH2)28. COOH.

1. Degeneration of the germinal epithelium of the testis invariably follows the operation of vasoligation or of vasectomy in the mouse, but does not become evident until a considerable period of time has elapsed.

2. Similar degeneration follows the operation of epididymo-deferentectomy, but after a much shorter interval of time.

3. This degeneration is caused perhaps by the pressure within the seminiferous tubules exerted by the accumulation of seminal products, or more probably by the action of toxic substances liberated from the disintegrating cell and sperm debris.

1. Experiments made at 15° C. with various adsorbents and gases (chiefly nitrogen and hydrogen) show that a cylinder filled with adsorbent granules has a capacity for dry gas under a given pressure which is generally greater than its capacity when containing no adsorbent. For example, a cylinder charged with nitrogen at 35 atmospheres has its capacity increased by 66 per cent. by filling it with cocoanut charcoal.

2. Sudden outbursts of firedamp in coal-mines are the result of releasing immense quantities of gas adsorbed under pressure in coal.

3. The logarithmic relation derived by Williams is shown to apply to the adsorption isotherms of gases above their critical temperature up to pressures of 100 atmospheres, providing a correction be applied for the gas in the capillaries which is not adsorbed, but which exists under simple compression.

The only observations, so far as I am aware, that have been made on the pitchers of Nepenthes seedlings, are those by Sir J. D. Hooker, who, in 1859, described their external configuration in his paper “On the Origin and Development of the Pitchers of Nepenthes, &c,” in vol. xxii. of the Linnean Society's Transactions, and afterwards added a few details of anatomical structure, in his admirable address on Insectivorous Plants, delivered at the Belfast meeting of the British Association in 1874.

In a previous paper the relation between the illumination I on a panchromatic plate, the duration t of the exposure, the wave-length λ of the light, and the resulting density D of the deposit on the developed plate was investigated.

Theories regarding the chemistry of the contraction process in striped muscle have been revolutionised during the last year. The Hill-Meyerhof theory was based on the assumption that the contraction process was due to the liberation of lactic acid by the break-down of glycogen, a non-oxidative reaction, and that the recovery process consisted in the oxidation of part of the lactic acid and the resynthesis of glycogen from the remainder. Meyerhof now believes (1, p. 305) that contraction is produced by the break-down of creatin-phosphoric acid (phosphagen), and that the energy produced by carbohydrate break-down is utilised to resynthesise phosphagen.

(1) Homarus vulgáris, Eupagurus bernhardus, and Carcinus mœnas all form limb-buds or papillæ in the process of limb regeneration. These are covered by a chitinous envelope, and the observations here recorded show that their outer form and size are adaptations to the requirements of the animal. That of the lobster is straight, that of the hermit crab curved, while the shore crab has a papilla which may be folded on itself three times inside the envelope.

(2) Valvular action of the diaphragm at the breaking plane plays a greater part in the stopping of haemorrhage after self-amputation than clotting, and the dilatation of small vessels which pass beneath the epidermis detaches a layer of cells. This layer of epidermis proliferates from its free edges to form the new limb.

(3) A new diaphragm is the first structure laid down, and differentiation takes place from the base outwards. Muscle arises at the growing tip from cells proliferated from the old epidermis (an ectodermal structure), and the nerve grows outwards from the torn end by cell proliferation.

(4) Muscle-fibres are anatomically complete immediately before moulting. The fibrillæ are cross-striated and enclosed in a sarcolemma, but full functional activity does not come till several days after moulting, beginning with slow rhythmic movements. Sarcoplasm seems to be less plentiful than in the normal fibre.

(5) When moulting occurs the papilla is at once expanded to several times its previous size by valvular action, and the epidermis, previously composed of several layers of cells, now thins to a single layer, as is seen in the normal limb.

Cyclic irrotational motion, [§ 60 (z)] once established through an aperture or apertures, in a movable solid immersed in a liquid, continues for ever after with circulation or circulations unchanged, [§60 (a)] however the solid be moved, or bent, and whatever influences experienced from other bodies. The solid, if rigid and left at rest, must clearly continue at rest relatively to the fluid surrounding it to an infinite distance, provided there be no other solid within an infinite distance from it. But if there be any other solid or solids at rest within any finite distance from the first, there will be mutual forces between them, which, if not balanced by proper application of force, will cause them to move.

This paper is the continuation of the papers by Born (1939, Part I) and by Born and myself (1940, Parts II and III). Here the theory is extended to the case of spinor wave functions satisfying the Dirac equation.

In a previous paper (Jack 1970, § 5) the pre-zonal polynomials were defined by the surface integral (1) below. In that paper they were evaluated using a series of differential type operators acting on the sums of powers of the roots of a matrix.

The present paper, although it is a continuation of Jack (1970), is self-contained and consists of three parts. The first part, §§ 3–7, shows how the pre-zonal polynomials may be calculated, certainly for low degree, using only the first of the differential operators. The next part, §§ 8–11, is devoted to establishing various recurrence relations between the pre-zonal polynomials and to conjecturing similar recurrence relations between the zonal polynomials (James 1964, p. 478). In the final part, §§ 12–14 the pre-zonal polynomials are evaluated directly from the integral (1) below, in terms of the augmented monomial symmetric functions.