The operators Δhf ≡ f(x) on function spaces and Δxn ≡ xn+1–xn on sequence spaces replace derivatives to yield analogues of the Kolmogorov inequality. Estimates for best constants are given for many spaces and for a few the best constants are actually given.

Using the KOℝ/2-theoretic obstruction theory developed in [4] and [5], necessary and sufficient conditions are derived for quaternionic projective spaces ℍPk and odd-dimensional complex projective spaces ℂP2k+1, of real dimension m say, to immerse in Euclidean space ℝ2m−1 in the range l ≦ 14. The results refine those obtained by Davis and Mahowald ([10, 11]) and earlier authors.

We discuss some existence theorems for partial differential inclusions, subject to Dirichlet boundary conditions, of the form where Φ is a quasi-affine function and so, in particular, for Φ(Du) = det Du.

We then apply it to minimization problems of the form

Let U be a convex open set in a finite-dimensional commutative real algebra A. Consider A-differentiable functions f: U → A. When they are C2 as functions of their real variables, their A-derivatives are again A-differentiable, and they have second-order Taylor expansions. The real components of such functions then have second derivatives for which the A-multiplications are self-adjoint. When A is a Frobenius algebra, that condition (a system of second-order differential equations) actually forces a real function on U to be a component of some such f. If v is a function of n real variables, and M is a constant matrix, then the requirement that M∇(u) should equal ∇(w) for some w usually falls into this setting for a suitable A, and the quite special properties of such v, w can be deduced from known properties of A-differentiable functions.

The existence and multiplicity of T-periodic solutions to a class of differential equations with attractive singularities at the origin are investigated in the paper. The approach is based on a new method of construction of strict upper and lower functions. The multiplicity results of Ambrosetti–Prodi type are established using a priori estimates and certain properties of topological degree.

We provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

A general decomposition theorem that allows one to express uniquely arbitrary differential polynomials in one independent and one dependent variable as a combination of conservative, dissipative and higher order dissipative pieces is proved. The decomposition generalises the Rayleigh dissipation law for linear equations.

We discuss the dyadic John–Nirenberg space that is a generalization of functions of bounded mean oscillation. A John–Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John–Nirenberg space and provide a method to construct nontrivial functions in the dyadic John–Nirenberg space. Moreover, we prove that the John–Nirenberg space is complete. Several open problems are also discussed.

We study the Benney equation and show that the associated initial-value problem is locally well-posed in Sobolev spaces Hs(ℝ2) for s > −2. Furthermore, we use a priori estimates to establish the global well-posedness for s ≥ 0. We also prove that these results are in some sense sharp. In addition, we obtain some exact travelling-wave solutions of the equation.

Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.

A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which continuous and discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel is constant, the fragmentation-rate function is linearly bounded, and the continuous mass-loss-rate function is locally Lipschitz, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is binary with constant fragmentation kernel and constant continuous mass loss, an explicit formula is given for the associated substochastic semigroup.

According to Percy's definition, “clays are termed fireclays or refractory clays when they resist exposure to a high temperature without melting or becoming in a sensible degree soft and pasty.” The refractoriness of fireclay is due to its low proportions of fluxes. The best known British fireclays occur beneath the coal seams of the coal measures, and are therefore known as “under-clays” or “seat-clays.” The explanation of their paucity of fluxes usually offered is that the alkalies have been withdrawn as food by the vegetation which formed the coal. Thus Professor Tarr remarks that fireclays “are particularly abundant in the Carboniferous rocks associated with coal beds, the plants having been instrumental in the withdrawal of the alkalies.” This view has been widely accepted, and the death of the successive coal forests has been attributed to the exhaustion of the plant foods in the underlying soils. Caution is, however, necessary in the application of this theory; for plants can only withdraw soluble alkalies from the soil, and the alkalies are usually present in the form of silicates, which decompose slowly. Hence the total removal of alkalies from a soil by vegetation would be a very lengthy process. Moreover, alkalies are not the only, or indeed the most common of fluxes, for iron is usually the most potent. Further, some good fireclays are not associated with coal seams, while well-washed river muds may be poorer in alkali than an underclay, the alkalies having been washed out of the material. Thus the river muds from the Rhine near Bonn, analysed by Bischoff, contain only ·89 per cent. of potash and ·39 per cent. of soda; or after deducting the water and organic material the percentages of potash and soda are 1·02 and 0·45 respectively. Another analysis by Bischoff of river mud from the Rhine collected above Lake Constance contained potash ·55 per cent. and soda ·54 per cent., or after deducting the water and carbonates the percentage of these two constituents only rose to ·91 per cent. of potash and ·90 per cent. of soda.

The Glossopharyngeal.—The glossopharyngeal is usually looked upon as the most typical of the cranial nerves. In Amia, according to Allis, it consists of post- and præ-branchial branches, a visceral or pharyngeal branch, and a dorsal branch which takes part in innervating the lateral line system—supplying by a single twig one of the sense organs of the lateral canal, and in addition a row of pit organs.

According to a usage which has been adopted from the German of Helmholtz by the best English scientific writers on sound, a sound is called a “simple tone,” or without qualification a “tone,” when the variation of pressure of the air in the neighbourhood of the ear which is the immediate excitant of the sense is according to a simple harmonic function of the time; that is to say, when the whole pressure of the air varies in simple proportion to the distance, from a fixed plane, of a point moving uniformly in a circle.

The volume marked “Logarithms 0” now placed on the Society's table contains, to 28 places, the Logarithms of all numbers up to Ten Thousand. In preparing it, care was taken that each prime should be placed in connection with two distinct sets of other prime numbers, in order that all likelihood of mistake should be avoided. This amounts to the computation of each logarithm by two independent processes. Opposite each prime, reference is made to the page of the record of calculations; and the accompanying volumes marked “Construction, I., II.” contain minutes of each articulate step, so that the genesis of any logarithm may readily be traced and examined.

We develop a technique for calculating the Wielandt subgroup of a semidirect product of two finite groups of coprime order. We apply this technique to calculate the Wielandt length of a supersoluble group in terms of the Wielandt lengths of its Sylow subgroups (for small Wielandt lengths) and in terms of the nilpotency classes of its Sylow subgroups.

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.

In this paper, we consider an n-dimensional semilinear equation of parabolic type with a discontinuous source term arising from combustion theory. We prove local existence for a classical solution having a ‘regular’ free boundary. In this regard, the free boundary is a surface through which the discontinuous source term exhibits a switch-like behaviour. We specify conditions under which this solution and its free boundary are global in time; moreover, we exhibit a special domain for which, for t tending to infinity, such a global-in-time solution converges, together with its free boundary, to the solution of the stationary problem and to its regular free boundary (which is proved to exist), respectively. We also prove uniqueness and continuous dependence theorems.

We prove that there are 22 non-isomorphic subdirectly irreducible double MS-algebras and give a complete description of them.

In a recent paper, Professor E. T. Whittaker discussed the problem of defining, in a general riemannian space-time, the concept of spatial distance between material particles. It is the object of this paper to give an alternative definition, and to compare the new formula with that of Whittaker.