We study a model of ferromagnetic material with hysteresis effects. The magnetic moment behaviour is described by the nonlinear Landau–Lifshitz equation with an additional term modelling the hysteresis. This term takes the form of a maximal monotone operator acting on the time derivative of the magnetic moment. In our model, it is approximated via a relaxing heat equation. For this relaxed model we prove local existence of regular solutions.

An initial-boundary-value problem is considered for the heat equation in an infinite angle dθr ⊆ R2 × [0, ∞) with the oblique derivative boundary conditions on the faces λi of the angle:

with either h0 + h1 > 0, or h0 + h1 ≦ 0. The unique solvability of such a problem is proved in appropriate weighted Sobolev spaces according to the sign of h0 + h1. Estimates of the solution are obtained under ‘natural’ restrictions on the opening of the angle.

We give a simple explicit description of the norm in the complex interpolation space (Cp[Lp(M)], Rp[Lp(M)])θ for any von Neumann algebra M and any 1 ≤ p ≤ ∞.

Louis Albert Necker, who died at Portree, in Skye, on the 20th November 1861, aged 76, had been for many years a Foreign Honorary Fellow of the Royal Society of Edinburgh.

His relation to the Royal Society, and to Scotland generally, was, however, far different from what belongs to most honorary members. As a youth, his studies had been pursued at the Edinburgh University. He had received almost his first introductions to society amongst the very best circles which the Scottish capital, in the days of perhaps its highest literary and scientific celebrity, could afford; he visited the Highlands, and even the remoter Hebrides, with an admiring enthusiasm which few native tourists have surpassed. In later life he returned with renewed interest to revisit the scenes where he spent his youth.

Hydroid Zoophytes were collected by the Nimrod at eleven different stations within the Antarctic Circle. These stations centre round three definite areas of Victoria Land—the open waters of M'Murdo Sound to the west and north-west of Cape Royds (“M'Murdo Sound” in the following pages); the open waters of the Sound from Cape Royds southwards (“Cape Royds” in the following pages); and the Bay, a small indentation in the land to the east of Cape Royds.

1. Among 7135 chick embryos of nine days or over, obtained from unhatched eggs from three poultry farms, 112 cases of chondrodystrophy were observed. From 1900 eggs set at this Department, 124 chondrodystrophic embryos were obtained.

2. The mortality caused by this condition varied in different flocks from 0.80 to 7.96 per cent. of the embryos dying after the eighth day; at this Department it was responsible for loss of 6.5 per cent. of the eggs set.

3 The incidence of chondrodystrophic embryos was found to be greatest in January and February, and to decline steadily thereafter to an almost complete absence in June.

4. The occurrence of chondrodystrophy appears to be independent of breed of fowl, sex of embryo, and age of dam.

5. The number of these abnormal embryos produced by individual hens ranged from 1 to 28.

6. Evidence is advanced in support of the theory that the causal agency is an hereditary physiological abnormality in the dam which, under certain unfavourable conditions, results in the production of chondrodystrophy in the embryo.

7. It is shown that the incidence of chondrodystrophy appears to be inversely proportional to the amount of sunshine, and the suggestion is made that lack of direct sunlight is a factor in the etiology of the abnormality.

The grey squirrel (Sciurus carolinensis leucotus Gapper), which is a native of the North American continent, was introduced into England and Scotland quite recently and soon became of great economic importance in these countries. In his very excellent monograph, Middleton (1930) states that both sexes of the grey squirrel show diminished activity during the winter and while they do not hibernate in the true sense of the word, they probably sleep if the weather is severe. The breeding habits of the female were studied by Deanesly and Parkes (1933), who ascertained that the female definitely has a period of anœstrus during which the reproductive organs are atrophic. In the male, however, fully functional testes were found throughout the year (Allanson, 1933), and there were no histological changes which would indicate a regular period of quiescence.

While the prismatic spectra of the blue portions of an oil-lamp or coal-gas flame, exhibit a number of bright lines, separated by dark intervals, the spectra derived from the bright light of these flames are perfectly continuous. Apparently inconsistent results are in like manner obtained when the flames of different compounds of carbon and hydrogen are compared. Thus, lines are easily seen in the spectrum of the flame of alcohol, which are invisible in that of the flame of oil of turpentine.

Let D be a compact convex planar domain containing the origin, the boundary of which is of class C∞ and has finite non-vanishing curvature throughout. For the number A(i) of lattice points in the “blown up” domain √tD, the estimate

is established. This is a generalization of Hardy's classical result for the circle problem. The proof is based on asymptotic formulae for certain exponential integrals due to E. Hlawka.

In this paper, I shall consider operators generated by difference equations of the form

where Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.

This paper summarises the results of an attempt to extend the theory upon which the relationship between linear differential equations and integral equations is based. The case in which the nucleus K(x, s) of the integral equation arises as a Green's function is well known; the nucleus is there characterised by its having discontinuous derivates when x = s. The method here dealt with is virtually an extension of Laplace's and analogous methods for solving linear differential equations by definite integrals, and leads to nuclei which are continuous and have continuous derivates for x = s.

We study the large-time behaviour of solutions to the Korteweg-de Vries-Burgers equation with bore-like initial data. This work relies on the methods of Amick, Bona and Schonbeck to obtain sharp rates of temporal decay of certain norms of the solution, thus obtaining an improvement over results of Naumkin and Shishmarev.

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku) dx on the space BVk(Ω:ℝm) of the integrable functions, whose the f-th derivative in the sense of distributions is a Radon measure with bounded total variation.

We discuss necessary and sufficient conditions for the existence of eigentuples λ=(λl,λ2) and eigenvectors x1≠0, x2≠0 for the problem Wr(λ)xr = 0, Wr(λ)≧0, (*), where Wr(λ)= Tr + λ1Vr2, r=1,2. Here Tr and Vrs are self-adjoint operators on separable Hilbert spaces Hr. We assume the Vrs to be bounded and the Tr bounded below with compact resolvent. Most of our conditions involve the cones

We obtain results under various conditions on the Tr, but the following is typical:

THEOREM. If (*) has a solution for all choices ofT1, T2then (a)0∉ V1UV2,(b)V1∩(—V2) =∅ and (c) V1⊂V2∪{0}, V2⊈V1∪{0}. Conversely, if (a) and (b) hold andV1⊈V2∪∩{0}, V2⊈ then (*) has a solution for all choices ofT1, T2.

This paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expression

on [0,∞), together with a real homogeneous boundary condition at t = 0.

Let n ≥ 0 be an integer and let λ(n) be the median of the Gamma distribution of order n + 1 with parameter 1. In 1986, Chen and Rubin conjectured that n ↦ λ (n) − n (n = 0, 1, 2, …) is decreasing. We prove the following monotonicity theorem, which settles this conjecture.

Let α and β be real numbers. The sequence n ↦ λ (n) – αn (n = 0, 1, 2, …) is strictly decreasing if and only if α; ≥ 1. And n ↦ λ(n) − βn (n = 0, 1, 2, …) is strictly increasing if and only if β < λ(1) − log 2 = 0.98519….

We prove that connected subsets of M2×2 without rank-one connections are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. Under a weaker condition that the set does not have rank-one connections locally, we are able to establish some global results on the set. We also establish some results on Lipschitz extensions of the functions thus obtained.

In this paper we prove existence of multiple positive solutions for a Neumann problem in ℝN/(0, R), R large, with a superquadratic and odd nonlinearity. The proof is based on the fact that in such a situation the minimum of the corresponding energy functional (which is achieved) is not an even function and that there is quite a large gap (for large R) between such a minimum and the minimum of the same functional on even functions. In the set of functions whose energy lies in such a gap, we can apply index theory to prove the desired multiplicity result.

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

The main purpose of this paper is to study the mean value properties of certain Hardy sums over a short interval by using the mean-value theorems of the Dirichlet $L$-functions. Our main result is a mean-value formula for these sums.