We study equilibrium surfaces for an energy which is a linear combination of the area and a second term which measures the bending and twisting of the boundary. Specifically, the twisting energy is given by the twisting of the Darboux frame. This energy is a modification of the Euler–Plateau functional considered by Giomi and Mahadevan (2012, Proc. R. Soc. A 468, 1851–1864), and a natural special case of the Kirchhoff–Plateau energy considered by Biria and Fried (2014, Proc. R. Soc. A 470, 20140368; 2015, Int. J. Eng. Sci. 94, 86–102).

This paper presents a systematic account of ℤ/2-equivariant KO-theoretic methods in the study of r-fields (that is, r linearly independent cross-sections, r ≧ 1) on a real vector bundle. Applications of the theory to the problem of immersing complex and quaternionic projective spaces in Euclidean space are given in [13].

Using some new expressions for the weakly convergent sequences coefficient WCS(X) the lower boundedness

is proved, where δ(-) is the (Clarkson) modulus of convexity. We also define a modulus of noncompact convexity concerning nearly uniformly convex spaces which is used to obtain another lower bound for WCS(X). The computation of this modulus in Ip-spaces shows that our second lower bound is the best possible in these spaces.

We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi =\{x=(x_{h}, x_3)\ \vert \vert x_{h} \vert \gt 1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi )$ satisfying the decay condition of the swirl component $ru^{\theta }_{0}\in L^{\infty }(\Pi )$.

We consider the parabolic one-dimensional Allen–Cahn equation

The steady state connects, as a ‘transition layer’, the stable phases –1 and +1. We construct a solution u with any given number k of transition layers between –1 and +1. Mainly they consist of k time-travelling copies of w, with each interface diverging as t → –∞. More precisely, we find

where the functions ξj (t) satisfy a first-order Toda-type system. They are given by

for certain explicit constants γjk.

We present a new method for solving the minimization problem in ferromagnetism. Our method is based on replacing the non-local non-convex total energy of magnetization by a new local non-convex energy of divergence-free fields. Such a general method works in all dimensions. However, for the two-dimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some recent non-convex minimization procedures in the calculus of variations. We focus on the two-dimensional case in this paper and leave the three-dimensional situation to future work. Special emphasis is placed on the analysis of the existence/non-existence depending on the applied field and the physical domain.

The present paper deals with non-real eigenvalues of singular indefinite Sturm–Liouville problems with limit-circle type endpoints. A priori bounds and the existence of non-real eigenvalues of the problem associated with a special separated boundary condition are obtained.

This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ ℝN. The inequalities bound (semi-)norms of the boundary trace by certain norms of the function, its gradient on the region and by two specific constants κρ and κΩ associated with the domain and a weight function, respectively. These inequalities are sharp in that there exist functions for which equality holds. Explicit inequalities in some special cases when the region is a ball, or the region between two balls, are evaluated.

Radial deformations of a ball composed of a nonlinear elastic material and corresponding to cavitation have been much studied. In this paper we use rescalings to show that each such deformation can be used to construct infinitely many non-symmetric singular weak solutions of the equations of nonlinear elasticity for the same displacement boundary-value problem. Surprisingly, this property appears to have been unnoticed in the literature to date.

In this study, we are concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes Korteweg equations of a compressible fluid in the half space. We assume that the space-asymptotic states and the boundary data satisfy some conditions so that the time-asymptotic state of this solution is a rarefaction wave. Then we show that the rarefaction wave is non-linearly stable, as time goes to infinity, provided that the strength of the wave is weak and the initial perturbation is small. The proof is mainly based on $L^{2}$-energy method and some time-decay estimates in $L^{p}$-norm for the smoothed rarefaction wave.

We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T: X → X on some Banach space X with the following property: T has a bounded H∞-functional calculus with respect to the unit disc 𝔻(that is, T is polynomially bounded) but T does not have any bounded H∞-functional calculus with respect to a Stolz domain of 𝔻 with vertex at 1. Also we show that for an R-Ritt operator the unconditional Ritt condition of Kalton and Portal is equivalent to the existence of a bounded H∞-functional calculus with respect to such a Stolz domain.

What appears to be an entirely new foot-print has lately been found in the red sandstone of this district. I have seen many of the foot-prints from the various quarries in the neighbourhood, but have not before observed this one, nor is it like any figured in Sir William Jardine's splendid work on the “Ichnology of Annandale.” The foot-prints in question were found in a bed about 20 feet from the surface, at Locharbrigg's Quarry, three miles from Dumfries. They exhibit the usual large hind and smaller fore foot; the impression of the hind foot measures ′·5 × 2′·6, the fore foot 2′·3 × 1′·9; the stride of the animal appears to have been about 10′. The impression of the hind foot does not interfere with that of the fore foot, as is the case with several of the foot-prints figured in Sir William Jardine's work, the interval between them being 2′: the hind foot, therefore, must have been put down in the rear of the fore foot when the animal was walking. The impression of the foot shows five toes, the thumb being placed far back. The most characteristic features in these foot-prints are the well-developed claws, and the oblique position of the toes, i.e., they are placed to march one behind the other. In almost all the foot-prints I have seen, where the toes can be made out, the middle one appears the most prominent; this foot-print is markedly distinct in this respect.

Following on the current observations made during 1908 in Loch Ness by the author and Mr W. Watson, a few observations were made in Loch Garry (Ness basin) during August and September 1909. Mr Macdonald, Fort Augustus, again assisted in the work. The Loch Ness observations showed that the currents in the loch were very confused, but certain general conclusions were drawn; e.g. (1) that when a lake has become stratified and the temperature discontinuity has appeared, the return current is nearly always found above the discontinuity; (2) that towards the windward end of the loch the return current may take place close to the surface; (3) that slow-currents are to be found below the discontinuity, and with the same direction as the surface current. All these conclusions are borne out by the Loch Garry observations, and some additional information has been obtained.

It is now well established experimentally that when an atom is caused to emit radiation of frequency v by collision with an electron, the amount U of the kinetic energy of the electron which is absorbed by the atom is given by the equation

where h denotes Planck's quantum of Action : an electron whose kinetic energy before the encounter is less than hv is incapable of stimulating the atom to emit the radiation, and is merely repelled from the atom without any loss of energy.

The lavas of the Scottish Carboniferous are predominantly basaltic. True andesites and rhyolites are conspicuous by their absence, whilst trachytes and allied rocks are present in quite subordinate quantity. This association of basalt and trachyte is in accordance with the fact that the basalts have a slight alkaline cast, which is evidenced by an alkali content greater than that of the average basalt, and by the occasional presence of nepheline and analcite amongst their constituents. The transition from basalts to more acid types is accomplished, not by way of the andesites, but through the mugearites, volcanic rocks with chemical affinities to the essexites. All transitions from the more felspathic types of basalts, the Jedburgh and Markle types, can be traced through mugearites to trachytes or allied rocks.

In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.

We study the boundary-value problem, for (λ/k,ψ),

Here ∆ denotes the Laplacian, H is the Heaviside step function and one of A or k is a given positive constant. We define

and usually omit the subscript. Throughout we are interested in solutions with ψ>0 in Ω and hence with λ/=0.

In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψ tendsto zero when the area of Aψ, tends to zero.

For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.

In this paper, we deal with the nonlinear Schrödinger–Poisson system

where λ > 0, V and Q are radial functions, which can be vanishing or coercive at ∞. With assumptions on f just in a neighbourhood of the origin, existence and multiplicity of non-trivial radial solutions are obtained via variational methods. In particular, if f is sublinear and odd near the origin, we obtain infinitely many solutions of (SP)λ for any λ < 0.

The body-temperature of the following fishes, crustaceans, and echinoderms has been examined and compared with the temperature of the water in which they live:—Cod-fish (Gadus morrhua), ling (Molva vulgaris), torsk (Brosmius brosme), coal-fish or saithe (Gadus virens), haddock (Gadus œgelfinus), flounder (Pleuronectes flesus), smelt (Osmerus eperlanus), dog-fish (Scyllium catulus), shore crab (Carcinus mœnas), edible crab (Cancer pagurus), lobster (Homarus vulgaris), sea-urchin (Echinus esculentus), and starfish (Asterias rubens). The minimum, maximum, and mean temperature difference for each species are given in the following table:—

The excess of temperature is most evident in the larger specimens. This is well shown in the case of the coal-fish, where in the adult it was 0°·7 C., and in the great majority (11 out of 12) of the young of the first year, 0°·0 C. The body-weight and the conditions under which the fish are captured probably form the most important factors in determining the temperature difference.

In 14 codfish, where the rectal, blood, and muscle temperatures were recorded in the same individual, it was found to be highest in the muscle and lowest in the rectum, the mean temperature difference being 0°·46 C. for the muscle, 0°·41 C for the blood, and 0°·36 C. for the rectum.

The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.