The tow-nettings on which this survey is based were taken mostly from Keppel Pier. The tide as a rule runs down the channel strongly for several hours after high water, and the current is sufficiently strong to hold out the tow-nets more or less horizontal. Owing to some peculiarity in the configuration of the sea-floor at Keppel, the downward current is much the stronger and often begins an hour or several hours before high water. The flood-tide is never strong enough nor constant enough to hold out a tow-net for any length of time, and so most of the observations are based on tow-nettings taken on the ebb. Tow-nettings were taken almost every day and the tow-nets were usually left out for an hour.

The problem of fitting a polynomial to data by Least Squares has engaged the attention of many writers. The methods of approach have been many and various. Continued fractions, determinants, the calculus of finite differences and sums, the method of moments, the linear combination of data, the use of the orthogonal polynomials of Legendre and Tchebychef, these and doubtless other instruments of analysis have been pressed into service. At the end of the present paper is given a selective bibliography, which we hope on a future occasion to complete and to supplement by adding brief indications of the standpoint and achievement of each investigator.

Techniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a two-parameter system of ordinary differential equations of the second order.

We show that the pseudovariety generated by all semigroups of order-preserving transformations of a finitechain has no finite pseudoidentity basis.

1. When a suspension of a large number of small rigid spheres in a viscous fluid is allowed to sediment under the action of gravity the rate of fall of any of the particles in the body of the suspension is

U = U0(1 – κΦ), where U0 is the rate of fall of an isolated single particle calculated according to Stokes' law , where α is the radius of the particle, η the viscosity of the fluid, ρ the density of the particle, σ the density of the fluid, and g the constant of gravity. Φ is the volume of suspended material per unit volume of the suspension, and κ is a constant whose value according to theory is 5·5 + c, where c is probably in the neighbourhood of 1·6.

In this paper, we study the decomposition of the Nehari manifold by exploiting the combination of concave and convex nonlinearities. The result is subsequently used, in conjunction with the Ljusternik–Schnirelmann category and variational methods, to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical Sobolev exponent.

We study the asymptotic behaviour, for a sequence of varying open sets Ωn, of the solutions un of nonlinear Dirichlet problems for a monotone Leray–Lions operator. The method is based on the comparison between the gradient of un and the corrector for the p-Laplacian corresponding to the same geometry as the monotone operator. The representation of the limit problem and a corrector result are obtained.

We prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.

In this paper we give a sufficient condition which guarantees that the zero solution of a population growth equation is uniformly stable.

Let T be the formally self-adjoint second-order elliptic differential expression

in ℝn, where the coefficients bj, ajk, q are real-valued and ajk ≈ akj. In this paper sufficient conditions for all positive integer powers Tm of T to be essentially self-adjoint on are obtained.

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

This paper is devoted to a study of differential equations and inequalities of the form

and

The results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε > 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.

1. This paper is the continuation of a former one (Erdélyi, 1939), and deals with the integration of the system of two partial linear differential equations of the second order

The former paper will be referred to as I; all the notations of I will be retained without further explanation.

This paper considers autonomous parabolic equations which have a homoclinic orbit to an isolated equilibrium point. We study these systems under autonomous perturbations. Firstly we prove that the perturbation under which the homoclinicorbit persists forms a submanifold of codimension one. Then, if a perturbation of this manifold is considered, we prove that a unique stable periodic orbit arises from the homoclinic orbit under certain conditions for the eigenvalues of thesaddle point.

We study a Stefan problem for coarsening of many small particles of solid phase in undercooled liquid phase which includes surface tension and kinetic undercooling effects. A mean-field model is derived by homogenization of the free boundary problem. It extends the classical theory of Lifshitz et al. by taking into account effects due to kinetic undercooling.

We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

In this paper we analyse the methodology of the theory of differential inclusions. First, we emphasize that any sequence of piecewise affine functions with successive elements obtained by perturbations of preceding functions in the sets of their affinity converges strongly, together with the gradients. This gives a simple algorithm with which to construct sequences of approximate solutions that converge to exact solutions (neither the specific choice suggested by ‘the method of convex integration for Lipschitz functions’ nor Baire category methodology is required). We then suggest a functional that is defined in the set of admissible functions and measures maximal oscillations produced by sequences of admissible functions weakly convergent to a given function. This functional can be used to prove that the set of stable solutions is dense in the weak topology in the closure of the set of admissible functions either via the Baire category lemma or via a specific choice of strictly convergent sequences.

We explain how the above-mentioned methods of finding solutions to differential inclusions are connected to earlier results on weak–strong convergence, i.e. to results on stability, in the calculus of variations and in differential inclusions.

We also include information on developments in the subject in the three years after the results of this work were obtained.

This paper is concerned with the effect of the delays on the bounded solutions of the nth order n ≧ 1) differential equation

Sufficient conditions involving the retardations τJ(j = 1,2, …, m) which insure that every bounded solution of the considered equation is oscillatory are given. The results obtained generalise recent ones in [3 and 4].