Since the discovery by Heusler in 1903 of a magnetic alloy of copper, manganese, and aluminium, considerable interest has been aroused by the peculiarities of its properties and the difficulty of their explanation. In the summer of 1905 a preliminary investigation of a sample of the alloy was made in the Physical Laboratory of Glasgow University, and the results obtained have been published in a paper by Professor A. Gray.

In this paper we discuss the problem

We show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

In August 1901 Mr Thomas Anderson, merchant, Hillswick, to whom and to other members of whose family I have on several occasions been indebted for specimens to illustrate the Zoology of Shetland, wrote to tell me that a dead whale had been found floating near Hillswick and had been towed into Roeness Voe. It was within three miles of land when seen by the fishermen and was claimed by the Crown. From the appearance of the animal Mr Anderson thought that it was a sperm whale, and he sent me the following description, from which it was clear that he had correctly recognised it.

The interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

We establish that if Ω is an open square and if P is an arbitrary point of Ω, then the solutions of the wave equation with Dirichlet boundary condition utt − Δu = 0 in R × Ω, u = 0 on R × Γ can remain strictly positive during an arbitrarily large time. In fact stronger results are proved, considering several points of Ω simultaneously and prescribing different, more general, behaviour of the solution at these points.

In this paper, we consider the equation

where A is a linear operator, N = ψ′ with ψ ∈ C1(E, R), and E is an Hilbert space. We suppose that N has a derivative at infinity N′(∞) and that 0 belongs to the spectrum of A–N′(∞). We prove an abstract theorem for multiplicity of solutions for the above equation. We then apply this theorem to the study of periodic solutions of Hamiltonian systems and of semilinear wave equations when the period is prescribed.

The genus Branchiomaldane, represented by a single species vincenti, was founded by Langerhans to contain certain small Polychætes, which he discovered living in sand-covered tubes, among algæ, on the northern shore of Teneriffe. Professor Mesnil has since found specimens, inhabiting mucous tubes situated on the lower side of the calcareous alga Lithothamnion, in rock-pools at St Martin, near Cape la Hague, and has given an account of some of the structural features of these worms, which he showed to be adult and hermaphrodite. Professor Fauvel considered B. vincenti to be a dwarf Arenicola, arrested in development; he therefore merged Branchiomaldane with Arenicola, and designated the worm Arenicola vincenti. The writer has investigated the anatomy of several specimens of this worm, kindly sent to him by Professor Mesnil, and has concluded that the genus Branchiomaldane should be retained; the original name of the worm—B. vincenti—is therefore employed in the following account.

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$. We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$, $s \in \mathbb {R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$.

Conditions are given which ensure that a weighted 2nth order ordinary differential equation on a half-line satisfy the strong limit-point and Dirichlet conditions. Perturbation terms are permitted which either satisfy certain pointwise bounds or integral type bounds. Not all of the coefficients of the equation are required to be non-negative.

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

The interpretation of optical interference experiments at low light intensities and with independent light sources is discussed. A new experiment on interference between modulated light beams is described briefly, and its significance is indicated. There are two appendices, one a fuller discussion of the conditions under which interference and intensity correlation may be produced, the other a comment on one aspect of Jánossy and Nagy's discussion (1956) of the feasibility of the alternating slit interference experiment.

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up (p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up . Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x 1 only, and W satisfies

Let be the one-sided maximal operator and let Ф be a convex non-decreasing function on (0, ∞), Ф(0) = 0. We present necessary and sufficient conditions on a couple of weight functions (σ, ϱ) such that the integral inqualities of weak type

and of extra-weak type

hold. Our proofs do not refer to the theory of Orlicz spaces.

In this paper it is proposed to study the problem of the vibrations of a uniformly-stretched membrane whose superficial density σ depends upon the distance r from a fixed point in the membrane in accordance with the law

in which C and a are constants. The constant a will be known as the critical radius and the circle r = a as the critical circle.

The main aim of this paper is to introduce an appropriate dyadic one-sided maximal operator , smaller than the one-sided Hardy–Littlewood maximal operator M+ but such that it controls M+ in a similar way to how the usual dyadic maximal operator controls the Hardy-Littlewood maximal operator. We characterize the weighted inequalities for this dyadic one-sided maximal operator.

Monthly observations, extending over one year, were made on the rectal temperature of 114 domestic fowls (Gallus gallus, ♀) and records from forty-one of these were obtained for two years. Six different breeds were used, each located in a separate pen, all under similar conditions, and the mean temperatures for each group were plotted out to form an annual temperature curve. It was found that—

1. The lowest temperatures occur in December, January, and February, and the highest in June, July, and August, corresponding in a general way with the temperature of the external air.

2. Barometric pressure does not appear to have any influence on the body temperature of the hen.

3. The curve of egg-production does not coincide with the annual temperature curve, the former reaching its highest level in April and May, the latter in June, July, and August.

If we compare the mean rectal temperature at two periods of the year when the external or weather conditions are approximately the same (April-May and September-October), but when the vitality of the birds, as indicated by the curve of egg-production, moulting, etc., is at a maximum and minimum respectively, we find that the figures are practically identical. This would seem to show that cyclical bodily changes have little effect on body temperature as compared with outside influences.

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—

Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

Let W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.

This paper is concerned with the following nonlinear Schrödinger system in ${\mathbb R}^3$

$$\left\{ {\beging{matrix}{ {-\Delta u + (1 + \alpha P(x))u = \mu u^3 + \beta uv^2,} \hfill & {x\in {\open R}^3,} \hfill \cr {-\Delta v + (1 + \alpha Q(x))u = \nu v^3 + \beta u^2v,} \hfill & {x\in {\open R}^3,} \hfill \cr {u,v > 0,} \hfill & {x\in {\open R}^3,} \hfill \cr } } \right.$$

$\beta \in {\mathbb R}$

$\mu ,\nu $

We prove that, for any positive integer k > 1, there exists a suitable range of α such that the above problem has a non-radial positive solution with exactly k maximum points which tend to infinity as $\alpha \to +\infty $ (or $0^+$). Moreover, we also construct prescribed number of sign-changing solutions.