We study the structure of symmetric vortices in a Ginzburg–Landau model based on Zhang's SO(5) theory of high-temperature superconductivity and antiferromagnetism. We consider both a full Ginzburg–Landau theory (with Ginzburg–Landau scaling parameter κ < ∞) and a κ → ∞ limiting model. In all cases we find that the usual superconducting vortices (with normal phase in the central core region) become unstable (not energy minimizing) when the chemical potential crosses a threshold level, giving rise to a new vortex profile with antiferromagnetic ordering in the core region. We show that this phase transition in the cores is due to a bifurcation from a simple eigenvalue of the linearized equations. In the limiting large-κ model, we prove that the antiferromagnetic core solutions are always non-degenerate local energy minimizers and prove an exact multiplicity result for physically relevant solutions.

In this paper we study the existence of solutions for a free boundary problem arising in the study of the equilibrium of a plasma confined in a tokamak:

where p > 2, Ω is a bounded domain in ℝ2, n is the outward unit normal of ∂Ω, α is an unprescribed constant and I is a given positive constant. The set Ω+ = {x ∊ Ω: u(x) > 0} is called a plasma set. Under the condition that the homology of Ω is non-trivial, we show that for any given integer k ≥ 1 there exists λk > 0 such that for λ > λk the problem above has a solution with a plasma set consisting of k components.

We introduce the notion of the boundary flow for minimal surfaces in Rn with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence on the initial data.

A highly sensitive method of measuring alternating currents of the order of microamperes and of any frequency is described. The current to be measured is passed through the filament of a diode valve. The main heating current of this valve is supplied by an independent oscillator working at the frequency of the current under measurement. The change in the plate current of the valve is measured, thus indicating the amplitude of the alternating current being measured. Theoretical and practical details of the methods of working are given and particulars of the method by which the calibration was tested. It is suggested that the method is particularly applicable to the determination of electro-magnetic field strengths by measuring the currents induced in an aerial.

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in ℝn × [0, ∞)\{(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ = 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫Rnu(·, t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫Rnu∞(·, t) → ∞ as t ↘ 0.

In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u(c2) when c1 < c2, and u(c) ↗ u∞ as c ↗ ∞.

Furthermore, u∞ is self-similar in the sense that u∞(x, t) = t−αw(|x| t−αβ) for α = 1/(q − 1), β = ½(q − m), and some smooth function w defined on [0, ∞), so that is a finite positive constant independent of t > 0.

An initial–boundary-value problem for a class of wave equations with nonlinear damping and source terms in a bounded domain is considered. We establish the non-existence result of global solutions with the initial energy controlled above by a critical value via the method introduced in a work by Autuori et al. in 2010. This improves the 2009 result of Liu and Wang.

We have ventured to bring our examination of this specimen under the notice of the Society, not so much for its pathological interest, as for the light which it throws upon the normal mode of growth of the skeleton. There has been a tendency of late to regard endochondral ossification as quite secondary in importance to that of membranous ossification, but we believe that the case illustrates in a very striking manner the important part played by ossification in cartilage in the growth of the greater part of the skeleton.

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.

We prove existence and multiplicity of solutions for the problem

$$\left\{ {\matrix{ {\Delta ^2u + \lambda \Delta u = \vert u \vert ^{2*-2u},{\rm in }\Omega ,} \hfill \hfill \hfill \hfill \cr {u,-\Delta u > 0,\quad {\rm in}\;\Omega ,\quad u = \Delta u = 0,\quad {\rm on}\;\partial \Omega ,} \cr } } \right.$$

$\Omega \subset {\open R}^N$

$N \ges 5$

$\lambda >0$

$2^*=2N/(N-4)$

$W^{2,2}(\Omega )$

We show that in an infinite straight pipe of arbitrary (sufficiently smooth) cross-section, a generalized non-Newtonian liquid admits one and only one fully developed time-periodic flow (Womersley flow) when either the flow rate (problem 1) or the axial pressure gradient (problem 2) is prescribed in analogous time-periodic fashion. In addition, we show that the relevant solution depends continuously upon the data in appropriate norms. As is well known from the Newtonian counterpart of the problem, the latter is pivotal for the analysis of flow in a general unbounded pipe system with cylindrical outlets (Leray's problem). It is also worth remarking that problem 1 possesses an intrinsic interest from both mathematical and physical viewpoints, in that it constitutes a (nonlinear) inverse problem with a significant bearing on several applications, including blood flow modelling in large arteries.

Of a number of absorbing and emitting systems subjected to radiation, the proportions which suffer given sequences of absorptions and emissions are examined. The simple results which follow when the proportions so affected are small are applied to the photographic action on the assumption that in exposure the grain acts as a single system. They indicate that the grain requires two absorptions in succession to render it susceptible to “full” development, and three absorptions with one intermediate “emission” to render it susceptible to “partial” development, the terms “full” and “partial” being defined. These results are found to apply over the whole range of wave-length tested, from 4800 Å. to below 2000 Å.

A method for determining the boiling-points of solids as well as liquids, such that little material is required, and only constant, easily reproducible conditions are involved, is much needed. None of these qualities is possessed by the distilling flask method.

The apparatus suggested consists of a bulblet with a bent capillary not less than 1 mm. in diameter. This is attached to a thermometer and suspended in a beaker (fig. 1). The latter contains water (for substances boiling below 100°), sulphuric acid (up to 200°), melted paraffin (up to 280°), or a mixture of sodium and potassium nitrates (45·5, 54·5 parts) for use from 220° upwards. A stirrer is provided.

Chloroform has been prepared both from alcohol and from wood-spirit. The latter has been used for the sake of cheapness; but as it is a mixture of several liquids, all of which do not yield chloroform, it gives an impure product, in a proportion which varies much, but is always below that obtained from alcohol. There is therefore not only no advantage, but the contrary, in using woodspirit, which is not, after all, much cheaper than alcohol.

In February 1870, the author made a communication to this Society on the Kombe Arrow-Poison of Africa, a product of the Strophanthus hispidus plant. In that communication the nature of its action on the various structures of the body, and the chemical composition of the seeds of the plant, which are the most active part, were described. It was pointed out that the action is chiefly exerted upon the heart and upon the muscles of the body, and that the seeds contain a crystalline active principle of the nature of a glucoside, to which the name Strophanthin was given

Professor Tait has considered the question of finding the square root of a strain, or what is the same thing, that of a matrix of the third order— A mode of doing this is indicated in my “Memoir on the Theory of Matrices” (Phil. Trans., 1858, pp. 17–37), and it is interesting to work out the solution.

The notation and method will be understood from the simple case of a matrix of the second order. I write to denote the two equations, x1 = ax + by, y1 = cx + dy.

In the year 1871, I read before this Society a memoir on the Gravid Uterus of Orca gladiator, in which I discussed the placentation of the Cetacea. This memoir was published in the Transactions for that year. On the present occasion I purpose describing the placenta in a Cetacean genus in which it has not hitherto been examined.

In a paper read before the Royal Society of Edinburgh, in June 1905, on the Hydrodynamical Theory of Seiches, Professor Chrystal published a number of formulæ from which could be calculated the periods and positions of the nodes of seiches in lakes of various shapes.

The solutions of the most general problems, involving variations of the three dimensions, are there made to rest ultimately upon certain typical cases in which the element of breadth remains constant while the depth is some defined function of the length. In the paper referred to, formulae are given immediately applicable to lakes of uniform breadth, whose longitudinal sections include concave and convex parabolas, quartic curves, and rectilinear shapes.

During recent years it has become increasingly possible to interpret unusual genetic phenomena by reference to peculiarities in chromosome behaviour. Examples of such cytological explanation are found in the studies of triploidy by Bridges and Anderson (1924) and Redfield (1930, 1932), and of translocations by Dobzhansky and Sturtevant (1930), Rhoades (1931), and Dobzhansky (1932a). But, thus far, the phenomenon of complete linkage in the male of Drosophila has not been explained satisfactorily by direct observation of spermatogenesis. Doubtless this has been due partly to technical difficulties, and partly to the fact that chromosome behaviour is only now beginning to be understood. However, the first difficulties have been eliminated by Metz (1926), Guyenot and Naville (1929), and Huettner (1930); and the second by Belar (1928), and more recently by Darlington (1932b).

The paper is concerned with the estimation of the number of maximal-length genus n nonorientable Wicks forms for n → ∞. It is shown how to apply a graph theory for this problem. It is proved that if M(n) be the number of all nonequivalent nonorientable genus n Wicks forms of maximal length, then