We discuss the radially, symmetric solutions and the symmetry breaking of the equation Δu + 2δe −u = 0 in D and u + b(∂u/∂n) = 0 on ∂D, where D is the unit disk in ℝ2, δ >0 and b is a constant. We prove that for any b < 0, there exists > 0 such that there are exactly two radially symmetric solutions for δ ∊ (0, ), one for δ = and none for δ > δ*b. For , where m is a positive integer, there are (b), k = 1, …, m, such that the equation has symmetry breaking at δ*k (b) on the lower branch of radially symmetric solutions.

The differential equation

in N dimensions is considered, where q(x) is periodic. When N = 1, it is known that the conditional stability set coincides with the spectrum and that these also coincide with two other sets involving eigenvalues of associated eigenvalue problems. These results have been proved by means of the Floquet theory and the discriminant. Here an alternative proof is given which avoids the Floquet theory and which applies to the general case of N dimensions.

We study the asymptotic behaviour of a Hele-Shaw flow produced by the injection of fluid from a finite number of points at different speeds. We prove that, as time tends to infinity, the boundary of the fluid domain approaches the circle centred at the barycentre of the injection points with weights proportional to the injection rates. The distances from the barycentre to the boundary points are estimated from both above and below.

We shall prove multiplicity of non-trivial solutions for the following p-Laplacian problems

where f(x,u) is continuous, periodic in x and has superlinear growth in u.

A study is made of the minimum number of generators of the n-th direct power of certain 2-generator groups.

We make some remarks on the Euler–Lagrange equation of energy functional $I(u)=\int _\Omega f(\det Du)\,{\rm d}x,$ where $f\in C^1(\mathbb {R}).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $\Omega$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$ is not constant on $\Omega.$ We also prove some results concerning the homeomorphism solutions, non-quasimonotonicity and radial solutions, and finally we prove some stability results and discuss some related questions concerning certain approximate solutions in the 2-Dimensional cases.

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

We study the orbital stability of standing waves for the Klein–Gordon–Schrödinger system in two spatial dimensions. It is proved that the standing wave is stable if the frequency is sufficiently small. To prove this, we obtain the uniqueness of ground state and investigate the spectrum of the appropriate linearized operator by using the perturbation method developed by Genoud and Stuart and Lin and Wei. Then we apply to our system the general theory of Grillakis, Shatah and Strauss.

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for

is locally well posed in the largest space where the three conservations

can be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

When lecturing on Electrostatics at the Glasgow and West of Scotland Technical College, the author endeavoured to impress upon his students the identity of the laws of force for magnetism and electricity at rest. He thought it would further this idea if he could get pictures of electrostatic lines analogous to those obtained from magnets by means of iron filings, and an analysis of the difference of the conditions in the two cases led to the conclusion that there was no reason why this should not be done.

When we shower iron filings upon a glass plate, or a sheet of paper, under which a magnet is placed, unlike poles are induced at the ends of each filing, and it becomes for the time being a little compass needle. Since it is very short, the total action upon it will be a couple turning it into the direction of the force at its centre, unless it be close to the magnet, in which case there will be, in addition, a translation along a line of force. This latter effect causes the filings to crowd towards the edges of the magnet.

Now, in electrostatics we also have an inductive action, opposite charges being induced in the near and far ends of each particle as it comes into the field, but we have the important difference, that whereas the magnetic poles are attached to the filing and cannot be separated from it, the electric charges will leave a body if they have an opportunity of doing so.

Extreme supporters of the principle of relativity find no place for an æther. Thus Lord Haldane in discussing the theory of Einstein writes: “Space, as a physical thing with unvarying geometrical properties, is to be banished, just for the same sort of reasons as the æther was banished before it. Only observable things are to be recognised as real in the new system of modern physicists.” On the other hand, many followers of Faraday and Maxwell have regarded æther as the primary real substance; all mass, momentum, or energy being mass, momentum, or energy of the æther. Without entering into a discussion of the significance of physical “reality,” we may, with most physicists, agree to use the concept of æther as giving a model, inadequate though it is at present, for the interpretation of physical phenomena.

The following pages are intended merely as a preliminary notice of an investigation, which has occupied me for some time past, and which, though still too incomplete for publication in full, is sufficiently advanced to render obvious the general character of the results, although, from the extensive and elaborate nature of the inquiry, a very considerable time must elapse before it is complete in all the requisite details.

Bunsen's classical experiments on the absorption of gases by liquids show that the coefficient of absorption in water and alcohol between 0° and 20° diminishes as the temperature rises. Bohr and Bock found that at higher temperatures the coefficients of some gases (hydrogen and probably nitrogen) pass through a minimum, hydrogen in water at 60° C, nitrogen not far from 100° C. These results were not confirmed by Winkler, who concluded from his experiments that the coefficient approaches a smallest value asymptotically. Recently Estreicher, working with Professor Ramsay, found a minimum in the solubility for helium in water at 25° C.

The order Amphipoda, of the Crustacea Malacostraca, which consists of the 3 sub-orders Hyperiidea, Gammaridea, and Caprellidea, is highly important in the economics of the sea as a source of fish food. Hyperiids figure prominently in the diet of species which feed pelagically. Hardy (1924), for example, found them amongst the principal items in the food of East Anglian herring. Gammarids are preyed upon by bottom feeders in general. Todd (1905, 1907) and Blegvad (1916) record them in varying percentages from the stomachs of adult cod, haddock, whiting, plaice, dabs, flounders, sole, and skate. Their chief importance, however, would seem to be in the diet of fishes in post-larval and adolescent stages. Todd (1907, 1914) found Gammarids the main support of cod under 15 cm. and of plaice and dabs under 10 cm. Blegvad (1930) states that in Danish waters, plaice in their first year feed almost exclusively upon them. Clark (1922) found that in the early stages of 4 species of skate, Amphipods occurred in from 50 to 84 per cent of the stomachs examined, and Steven (1930) found the food of young stages of R. clavata and R. maculata to consist chiefly of Gammaridea. Gammarid species, as recorded by Hunt (1924), are also eaten by a variety of bottom-dwelling invertebrates.

IN 1903 Heusler made the important discovery that alloys of copper, manganese, and aluminium are strongly magnetic. It was shown in the paper published by Heusler that by properly adjusting the amounts of the component metals present an alloy is obtained the permeability of which is comparable with that of cast iron. Heusler also demonstrated the fact that the properties of this alloy are profoundly modified by thermal treatment.

After giving a summary of recent literature on the subject, the author then proceeded to state the nature of the material which he had employed in the present investigation.

The bird selected was the common fowl (Gallus domest.), and he had examined a series of microscopic sections through the limbs from the fourth day of incubation to the day of hatching.

The mammalian embryos examined were mice and rabbits, and the fingers of the human fœtus from an embryo approaching the full period of uterogestation.

The preparation of adamantine carbon or diamond has exercised the genius of philosophers from the very earliest times; but it was not until the middle of the last century (1772) that Lavoisier established the diamond's true nature—notwithstanding the simplicity of the experiments required to demonstrate the fact—and showed it to consist of pure carbon in a crystallised state. Since that time very many attempts have been made to prepare it artificially, but until the recent and now famous experiments of Mr J. B. Hannay there has not been the slightest approach towards the solution of this problem. Great obstacles stood in the way of success, the chief being the complete insolubility of carbon in all known liquids, coupled with its non-volatility and infusibility; while the subject was rendered even more difficult and obscure, by ignorance of the conditions under which the diamond is produced in nature, its peculiar crystalline form, together with extreme rarity, indicating a probable very slow formation, and rare natural existence of the conditions necessary for its formation.

I have given the name of Radiant Spectrum to a phenomenon which I discovered in 1814, and which I described to this Society in the early part of that year.

It will be understood from fig. 1, which represents the brilliant radiation which surrounds a very small image of the sun, when it is formed either by reflection or refraction, or otherwise.