The asymptotic behaviour of the solutions of Poincaré's functional equation f(λz) = p(f(z)) (λ > 1) for p a real polynomial of degree ≥ 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(zρF(logλz)), if f(z) → ∞ for z ∞ and z ∈ W, where F denotes a periodic function of period 1 and ρ = logλ deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

We give an interpretation of the Chern–Heinz inequalities for graphs in order to extend them to transversally oriented codimension one C2-foliations of Riemannian manifolds. It contains Salavessa's work on mean curvature of graphs and fully generalizes results of Barbosa–Kenmotsu–Oshikiri [3] and Barbosa–Gomes–Silveira [2] about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. This point of view of the Chern–Heinz inequalities can be applied to prove a Haymann–Makai–Osserman inequality (lower bounds of the fundamental tones of bounded open subsets Ω ⊂ ℝ2 in terms of its inradius) for embedded tubular neighbourhoods of simple curves of ℝn.

We show that a PD3-complex P such that π = π1(P) is infinite and has a non-trivial finite normal subgroup must be homotopy equivalent to RP2 × S1. Hence if A is an abelian normal subgroup of a 2-knot group πK which is not contained in the commutator subgroup πK′ and πK′ is infinite then A is torsion free.

1. Let (x1, x2, …, xn) be Cartesian coordinates in Rn. We identify the point X = (x1, x2, …, xn) with these coordinates with the vector X of components x1, x2, …, xn and use the usual vector notation.

The steady state theory of sound propagation in exponential loud-speaker horns is well known. The problem of transients, however, has not been treated hitherto, so in this paper it is proposed to consider what happens when the air particles at the throat of the horn are suddenly set in motion. To avoid difficulties which arise in treating sound propagation in the horn when the linear dimensions of the cross-section are comparable with the wave-length of the sound, the following

artifice is postulated. The axis of the horn is linear and the air column is divided up into a large number of frictionless conduits by very thin rigid partitions as indicated in Fig. 1. The cross-section of each conduit expands exponentially from the throat towards the mouth of the horn. The major linear dimension of the section of a conduit at the mouth is small in comparison with the wave-length of the highest frequency to be reproduced.

In this paper I consider the problem of integrating a generalized hypergeometric function with respect to one of its parameters, together with some general theorems on hyper-geometric transforms which arise out of this problem, and some interesting examples on these theorems.

The Brown–Grossman proper homotopy groups at an end of a locally compact Hausdorff space, X, were introduced by E. M. Brown in 1974. In this note we define and study ‘global’ versions of these groups and compare these global groups with the groups ‘at infinity’. We also obtain several interlocking exact sequences relating these groups with the Hurewicz and Steenrod homotopy groups of X.

Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

The probability of ever returning to the origin and the mean square displacement after n steps are studied for some lattice-valued random walks, whose successive steps constitute a Markov chain on a finite state space with transition probabilities of a simple kind, and such that the returns to the origin form a regenerative phenomenon. The case of walks on a diamond lattice with no immediate reversals is included: this example is relevant as a polymer chain building model. The numerical evaluation of the return probabilities of some three-dimensional walks is discussed and examples given.

Representations of the four-dimensional unitary group U4 were considered long ago by Wigner(1)as a model for nuclear isobaric multiplets. However, the Hamiltonian did not include the components of isospin which together with the spin coordinates are known to label the nuclear states. In this paper we shall find representations characterized by the eigenvalues of angular momentum J, isospin T and parity π, and will find mass relations which give good agreement with the experimental energy levels of Li6 and Be8 labelled by the same quantum numbers. The representations found by Wigner give good results for the ground state energies, or masses, of all the nuclei up to a mass number of A = 110(2), and we shall derive Wigner's representations as a special case. In fact, unless these are satisfied it is impossible for particle-like representations of U4 to exist!

In (2) we defined the Künneth suspension of a cohomology operation —the Künneth suspension involves an arbitrary css-complex Y rather than the 1-sphere S1, as with the usual suspension of a cohomology operation. Now the suspension homomorphism is well known to be related to the operation of forming loop spaces (cf. (4)). The main object of this paper is to prove a similar result for the Künneth suspension.

The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix ΘC. We develop and present a new technique to compute the expected value of tr(ΘCn) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [Rud10] and Chinis [Chi16]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [BDF+16] and [Zha]. We extend [BDF+16] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L-functions L(1/2 + it, χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.

In [5], we studied the behaviour of the Kähler cone of Calabi–Yau threefolds under deformations. We saw that the Kähler cone is locally constant in a smooth family of Calabi–Yau threefolds, unless some of the threefolds Xb contain elliptic ruled surfaces. Moreover, if X is a Calabi–Yau threefold containing an elliptic ruled surface, then the Kähler cone is not invariant under a generic small deformation.

Let G be a group. A subset D will be called a set of 3-transpositions if |x| = 2 for xεD and |xy| = 3 whenever x, yεD do not commute. We will call the set D closed if xDx−1 = D for each xεD. For each xεD, let

For each subset X of D, we denote by [X] the graph with vertex set X where two elements x, yεX are joined by an edge whenever they commute. We denote by (X) the complement graph; thus two elements x, yεX are joined by an edge of (X) whenever they do not commute.

The liquid drop model of an imperfect gas in the form introduced by Wergeland is discussed by using the method of the grand ensembles and the equation of state of the system is derived. This equation of state is of the same general type as the one derived by Mayer for a more general model. It is shown that in both cases the isotherms consist of two analytically different parts in the limit where the number of particles in the system, N, goes to infinity.

This paper is concerned with group presentations satisfying the small cancellation condition T(6). The definition of this condition is given in §1·2, together with some examples. Before giving the definition, however, we describe (in §1·1) some material which, to a certain extent, motivated our paper. In § 1·3 we state our main theorem, which provides new solutions to the word and conjugacy problems for T(6)-groups.

For proving the existence of periodic orbits of autonomous ordinary differential equations, three different methods are available, namely, the Hopf bifurcation theorem, the torus principle and the Poincaré–Bendixson theorem. Until recently, the Poincaré–Bendixson theorem was applicable only to equations of the second order. However, it was extended in (13, 14) to certain equations of higher order by means of a plane projection technique. In §§ 2, 3 of the present paper this technique is adapted to prove analogues of the Poincaré–Bendixson theorem for a class of autonomous retarded functional differential equations. For such equations the orbits lie in a Banach space studied by Hale ((7), p. 43). The main hypothesis of this theorem is that the equation has a bounded semi-orbit whose ω-limit set contains no critical points. The problem of finding such a semi-orbit is solved in § 4 for a class of dissipative equations. The practical application of these results requires the construction of certain functionals which are similar to the Liapunov functionals of stability theory. In § 5, these functionals are constructed for a large class of retarded feedback control equations and explicit conditions for the existence of periodic orbits are deduced. Some practical details are illustrated in § 6 by applying the theory to certain scalar delay-differential equations.

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd [3] have introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd [5] introduced alternate s-dimensional packing dimension profiles P-Dims by using Caratheodory-type packing measures, and proved, among many other things, that P-DimsE = DimsE for all integers s > 0 and all analytic sets E ⊆ RN.

The aim of this paper is to prove that P-DimsE = DimsE for all real numbers s > 0 and analytic sets E ⊆ RN. This answers a question of Howroyd [5, p. 159]. Our proof hinges on establishing a new property of fractional Brownian motion.

The object of this paper is to complete the proof of the

Theorem. Let P(√d) be the quadratic field of discriminant d > 0. Then Euclid's algorithm does not hold in P(√d) if d is sufficiently large.

The various processes occurring at the surface when hydrogen is adsorbed on tungsten are considered together with the dissociation equilibrium of hydrogen in the gas phase. The form of the adsorption isotherm is deduced from the principle of detailed balancing and is in agreement with that obtained by Fowler using a statistical method. A detailed interpretation of the experimental results now available shows that either (a) measurements of the rate of removal of the adsorbed film of oxygen on tungsten do not measure the rate of evaporation of oxygen atoms or (b) it is not possible to obtain a general first approximation formula giving the rate of evaporation of adsorbed atoms in terms of the heat of desorption. The desorption of hydrogen from tungsten is discussed and it is shown that the agreement between the temperature at which the film evaporates at an appreciable rate and that deduced from a desorption formula of the type mentioned in (b) assuming that the hydrogen evaporates as atoms must at present be regarded as a coincidence.