The H-spaces considered have no homology p-torsion and are rationally equivalent as H-spaces to products of even dimensional Eilenberg–Maclane spaces. We obtain conditions which ensure that if the cohomology with coefficients in the ring of integers localized at the prime p is a polynomial algebra, then the Pontrjagin ring with these same coefficients is polynomial. A topological consequence is that BSUP has just one homotopy associative, homotopy commutative H-structure.

We extend a result of J. F. Toland concerning bifurcation from infinity and we made some applications to variational inequalities.

Given a positive, decreasing sequence a, whose sum is L, we consider all the closed subsets of [0, L] such that the lengths of their complementary open intervals are in one-to-one correspondence with the sequence a. The aim of this paper is to investigate the possible values that Assouad-type dimensions can attain for this class of sets. In many cases, the set of attainable values is a closed interval whose endpoints we determine.

We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $\lambda > 0$, there is a travelling wave solution to fKdV and fNLS $\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $, which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in Hα/2[ − T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.

In this paper we prove, on one hand, extrapolation from infinity for the one-sided classes , and on the other hand, the BMO-boundedness for one-sided singular integrals. We also provide several applications of our results.

In the communications made by Sir Robert Christison to the Society in December and April last on the deep-water temperature of Loch Lomond, from observations made by him with a Miller-Casilla thermometer, these important facts were stated:—

(1.) On 12th October 1871, the temperature at the surface was 52·°0, from which it fell, on descending, till at 300 feet below the surface it stood at 42°·0, and this temperature of 42°·0 was uniformly maintained at greater depths or to 518 feet, the depth of the loch at the place of observation.

Among the undescribed species of plants occurring in the well-known material of Calciferous sandstone age found near Pettycur, Fife, is an interesting specimen belonging to the older type of Lepidodendroid structure which has a solid primary xylem.

Records relating to the occurrence of medusæ in the Firth of Clyde have not been numerous, though its waters have long been a noted resort for marine naturalists. Forbes, in his Monograph on the British Naked-eyed Medusæ, states that he paid special attention to the medusæ of the Clyde, and that those which he collected while on a yachting cruise in the autumn of 1839 were the foundation of his well-known monograph. Amongst those who have made additions to the list of species I may mention the names of Landsborough, Alder, Allman, Herdman, and two well-known foreigners, Claparède and Haeckel. In June 1902 Dr Clemens Hartlaub, of Heligoland, made a short stay at the Millport Biological Laboratory, solely for the purpose of collecting medusæ, but the complete results of his labours have not yet been published.

We consider some Bernoulli free boundary type problems for a general class of quasilinear elliptic (pseudomonotone) operators involving measures depending on unknown solutions. We treat those problems by applying the Ambrosetti–Rabinowitz minimax theorem to a sequence of approximate nonsingular problems and passing to the limit by some a priori estimates. We show, by means of some capacity results, that sometimes the measures are regular. Finally, we give some qualitative properties of the solutions and, for a special case, we construct a continuum of solutions.

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

Let

be Hermite's function of order λ and let h = h(λ) be the largest real zero of Hλ(t). Set

In this paper we establish the inequality

where

Equality holds for S = ½. The result is also fairly accurate as S→0 and S→1. The proof is analytical except in the ranges −1·1 ≦ h ≦ −0·1 and where the argument is concluded by means of a computer.

The following deduction is made elsewhere [2, Theorem A]. If u(x) is subharmonic in Rm(m ≧ 2) and the set E where u(x) > 0 has at least k components, where k ≧ 2, then the order ρ of u(x) is at least ϕ(1/k). In particular, if ρ < 1, E is connected. This result fails for ρ = 1.

In this paper we obtain necessary and sufficient conditions for the existence of Lipschitz minimisers of a functional of the type

where h is a convex function converging to infinity at zero and u is subjected to displacement boundary conditions. We provide examples of body forces f for which the infimum of J(.) is not attained.

The global solution, stability and evolution of regularities are studied for the initial-boundary-value problem of a viscous compressible gas flow. With global a priori estimates and careful analysis, the existence of global classical solutions in Hölder space is established.

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.

Bivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.

The diffraction of a line source by an absorbing semi-infinite plane in the presence of a subsonic fluid flow is examined. Expressions for the total far field for the leading edge (no wake present) and the trailing edge (wake present) situation are given.

It is found that the presence of fluid flow attenuates the sound level in the shadow region of the absorbent screen. There is greater attenuation of the sound level in the trailing edge situation than there is for the leading edge situation.

Sound level attenuation on the source side of the absorbent screen is found to depend more critically on the absorptive properties of the screen than on the fluid flow. The radiated sound intensity, in the half space in which the source is located, can be considerably reduced by a suitable choice of the absorption parameter.

It is established that under certain restrictions the solution u of the equation uxy − gu = 0 satisfying u(x, 0) = p(x) and u(0,y) = q(y) in ([0, = ∞) × [0, ∞), changes sign in where (X, Y) is any point in the relevant region.

Formulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

This paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) and

The extension consists in replacing f′ by M[f] where

choosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the form

where K is a positive number independent of f.

The analysis involves a fourth-order differential equation and the second-order equation associated with M.

A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

Non-negative solutions of the initial boundary value problem for a degenerate parabolic equation are investigated. It is shown that solutions blow up regionally in finite tine. The size of blow-up sets is determined for radially symmetric cases.