We consider the Sturm–Liouville equation with the initial condition and suppose that Weyl's limit-point case holds at infinity. Let ρα(μ) be the corresponding spectral function and its symmetric derivative. We show that for almost all μ ∈ R, if exists and is positive for some α ∈ [0, π), then (i) exists and is positive for all β ∈ [0, π), and (ii) for all α1, α2 ∈ (0, π) \ {½ π},

For a singularly perturbed nonlinear elliptic equation ε2Δu − V(x)u + up = 0, x ∈ ℝN, there are solutions concentrating around critical points of V with different energy scales with respect to small ε > 0. We combine the solutions when .

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

We discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

In this paper we deal with a singular elliptic problem involving an asymptotically linear nonlinearity and depending on two positive parameters. We investigate the existence, uniqueness and non-existence of the minima of the functional associated with the problem and, by employing a natural and very general definition of a weak solution, we also obtain a bifurcation-type result.

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

This subject is treated under eight different heads:—1. On the law of visible direction in Monocular vision. 2. On the law of visible direction in Binocular vision. 3. On vision of objects of three dimensions, by monocular vision; by binocular vision. 4. On the binocular vision of figures of different magnitudes. 5. On the cause of the perception of objects in relief by the coalescence of dissimilar pictures. 6. On the doctrine of corresponding points. 7. On the vision of cameos and intaglios. 8. On the change in the apparent position of the drawings of solid bodies.

1. Let ψ, φ … be the values at time t, of generalised co-ordinates fully specifying the positions of any number of solids movable through space occupied by a perfect liquid destitute of rotational motion, and not acted on by any force which could produce it. Some or all of these solids being perforated, let χ, χ′, χ″, &c., be the quantities of liquid which from any era of reckoning, up to the time t, have traversed the several apertures.

§ 1. Object of Paper.—The chief object of the present paper is to establish the following theorem:

The functions of Lamé [that is to say, the doubly-periodic solutions of the differential equation

are the solutions of the homogeneous integral-equation

where η denotes an arbitrary constant.

The view that a peculiar likeness exists between the northern and southern extra-tropical faunas, and particularly between those of the Arctic and Antarctic regions, was suggested by Théel in discussing the remarkable deep-sea group of the Elasipoda, whose discovery we owe to the Challenger Expedition. A somewhat similar view is hinted at or referred to more than once in other Reports of the same Expedition. It was afterwards stated in an ampler way by Pfeffer (Versuch über die Erdgeschichtliche Entwickelung der jetzigen Verbreitungsverhältnisse unserer Thierwelt, 1891), and has of late been dealt with in great detail, and in relation to the antecedent causes that might have led to such a phenomenon, by Sir John Murray. On the other hand, Dr Ortmann, considering the hypothesis from the point of view of our knowledge of the distribution of Crustacea, has rejected it entirely (“Uber Bipolarität in der Verbreitung mariner Thiere,” Zool. Jahrb., 1896; cf. also “Marine Organismen und ihre Existenzbedingungen,” ib., 1897), and Dr Chun, dealing with the pelagic fauna (“Die Beziehungen zwischen dem arktischen und antarktischen Plankton,” Stuttgart, 1897), while showing how in truth a certain small number of forms are common to far northern and far southern seas, holds that the facts are sufficiently accounted for by the continuous distribution or gradual intermixture of forms in the depths of the intervening oceans under present conditions, without our needing to have recourse to an explanation of the phenomenon in the different conditions of a former age.

The temperature of the air depends on so many varying factors that its prediction is a matter of considerable difficulty, and can only be made with any degree of certainty when the minimum number of these factors is at work. Underground temperatures are dependent not only on the same factors as affect the air temperature, but, in addition, are much more affected by conductivity, rainfall, evaporation, latent heat, etc., and thus the changes in temperature beneath the surface of the soil constitute a much more complex problem.

Coefficients of inbreeding have been calculated for 676 stallions and mares, with the object of studying the part played by inbreeding in the development of the Clydesdale breed of horses. Wright's coefficient of inbreeding was used in this study.

It is found that during the early history of the breed very little inbreeding had been practised, and it is suggested that this may be due to the fact that the foundation stock contained many recessive factors for undesirable characters. The first marked rise in the average percentage of inbreeding for the breed occurs during the period 1880–1890, and from that time onwards there is a gradual but constant increase in the breed average, reaching its highest level at 6.25 per cent. for the period 1920–1925.

A method is outlined by which a measure can be obtained of the contribution of any particular sire to the graph of the average percentage of inbreeding for the breed. It also measures the degree of concentration of his blood in animals inbred to him and the rate at which his blood is diffused through the breed.

Professor Tait has shown that accurate determinations of surface-tensions would be obtained if we could measure the rate of propagation of ripples set up by the vibration of a tuning-fork of known pitch.

The portable gold-leaf electrometer designed by C. T. R. Wilson gives a means of measuring the charge upon and current through a conductor exposed to the earth's field and maintained at zero potential. Further, Wilson has shown (1) that the dissipation factor (the ratio of current per minute to the corresponding charge) is approximately the same for a surface of turf and the metal test-plate; and (2) how to deduce the corresponding charge and current per square centimetre on the neighbouring ground-level. Wilson's measurements were made chiefly in a country atmosphere (near Peebles), and we have made observations with a similar instrument in town air, in and near Edinburgh, to find out whether the dissipation factor is notably affected by the purity of the atmosphere as regards smoke, etc.

This paper concludes the study of fitting polynomials by Least Squares, treated in two previous papers. The problem being concerned with the minimum of a positive definite quadratic form, it makes for conciseness to use matrix notation. We shall therefore adopt the following conventions :—

The n values of the variable x, of the data u0, u1, …, un−1, of certain polynomials qr(x) entering into the solution, and so on, will be regarded compositely as vectors. They will be imagined as having their components or elements disposed in column array, but when written in full will be written horizontally, to save space, enclosed by curled brackets. Row vectors, when written out in full, will be enclosed by square brackets. In the shorter notation we shall write, for example, u, x for column vectors, u′, x′ for the row vectors obtained by transposition. The vectors occurring in the problem will be the following:—

I was recently led to consider this subject by Professor A. Dickson, who showed me some of his beautifully-mounted specimens, and explained to me the method he employs for the determination of the divergence, and of the successive leaves of the fundamental spiral or spirals. He referred me to two terribly elaborate papers by Bravais, and I have since met with another of a similar character by Naumann. These papers certainly cannot be supposed to present the subject from the simplest point of view.

Vertebraria is one of the commonest and most characteristic fossils in those Upper Carboniferous and Permian strata in which Glossopteris is the dominant fossil plant. It has been found in the form of coaly incrustations, impressions, and casts, and accounts of these various forms have been given by Arber and Seward. In view of our very inadequate knowledge of Vertebraria and Glossopteris any information about them is of importance. The genus Vertebraria Royle was founded on impressions of plant axes, some of which, from the evidence of the attachment of roots, must be judged to be roots or rhizomes. Vertebraria axes are usually found lying in the plane of the bedding of the matrix (Pl. I, fig. 2) and less frequently at right angles to the plane of bedding (PL I, fig. 3).

During the first seven months of the year 1908 I had the good fortune to be living at Invergarry in Inverness-shire, with sufficient time at my disposal to make temperature observations in Loch Garry, and, through the generosity of Sir John Murray and Mr Laurence Pullar, the funds of the Lake Survey (Pullar Trust) were put at my disposal to defray the expenses of observation. I was further fortunate in securing the services of Mr Wm. Macdonald, Mount Pleasant, Fort Augustus, who was previously boatman to the Lake Survey on Loch Ness, and who proved himself a most painstaking and eager observer, and by whom by far the greater number of the actual observations were made.

On the Mechanical Action of Radiant Heat or Light.

It is assumed in this communication that the undulatory theory of radiant heat and light, according to which light is merely radiant heat, of which the vibrations are performed in periods between certain limits of duration, is true. “The chemical rays,” beyond the violet end of the spectrum, consist of undulations of which the full vibrations are executed in periods shorter than those of the extreme visible violet light, or than about the eight hundred million millionth

In the course of a recent investigation on the fertilised egg of the mouse (Gresson, 1940 a, b) it was necessary to keep a record of the time at which pairing took place. This revealed that copulation, especially in the winter, was very infrequent during the hours of daylight, and usually took place at about 12 midnight or between midnight and early morning. Similar observations were made by Lewis and Wright (1935), who found that in the autumn and winter mice seldom copulated during the day, and that while the majority of pairings took place between 10 p.m. and 1 a.m., several of their mice paired between 1 a.m. and 7 a.m., whereas in the spring pairing occurred more readily during all hours of the day.