It is shown that the spectrum of a linear operator T on a complete linear metric space X is a Borel set. If, in addition, X is locally convex or separable, then the spectrum of T is a Gδσ set.

We study the right-definite separated half-linear Sturm–Liouville eigenvalue problems. It is proved that the $n$th real eigenvalue of the problem depends smoothly on the equation, but may have jump discontinuities with respect to the boundary condition. Formulae are found for the derivatives of the $n$th real eigenvalue with respect to all parameters: the endpoints, the boundary condition and the coefficient functions, whenever they exist. Monotone properties and a comparison result for real eigenvalues are deduced as consequences. The generalized Prüfer transformation and the implicit function theorem in Banach spaces play key roles in the proofs.

In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q > n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q > n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝd) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For α > 0 this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for α < 0 it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

Throughout this article, the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. We construct an ideal called the socle-ideal of N which is antiradical in the sense that it is a direct sum of minimal left ideals and annihilates one or more of the radicals of N. We then use this socle-ideal to obtain a decomposition theorem for the s-radical JS(N) of N.

This paper investigates the effect on the structure of a band of imposing conditions on the congruence or right congruence lattice of the band. Bands whose congruence lattices are distributive or Boolean are characterized, as are bands whose right congruence lattices are Boolean.

This paper studies the boundary-value problem arising from the behaviour of a fluid occupying the region 0 ≦ x ≧ 1 between two rotating discs, rotating about a common axis perpendicular to their planes, when the discs, are rotating in the same sense with speeds 0 ≦ Ω0<Ω1. The equations which describe the axially symmetric similarity solutions of this problem are

with the boundary conditions

where ε = v/2Ω1 and v is the kinematic viscosity.

The major result is: There is an ε0 such that for 0<ε≦ε0 there does not exist a solution 〈H(x,ε), G(x,ε)〉 with G′(x,ε)≧0.

For a series of elements of a topological vector space, necessary and sufficient conditions are found, in terms of the set of finite partial sums, for unconditional convergence and for the corresponding Cauchy condition. The extent to which these results remain valid for topological groups is investigated. A new and direct proof, for locally convex spaces, is given of the theorem of Orlicz.

The author showed to the Society a small entire specimen of the P. gladius, and next described, from a larger opened and dissected one, and from part of an adult fish, the spinal cord, the brain, the organs of the senses, and other parts of its nervous system. He illustrated his remarks by exhibiting four large drawings and nine smaller ones, including six microscopic views, explanatory of his description of the structure and disposition of the spino-cerebral axis, the encephalon as viewed from above and below, the ramifications of the encephalic nerves, and more particularly the structures subserving the senses of smell, sight, and hearing.

The chemistry of this not the least interesting of deep-sea deposits, though it has received attention from several investigators in the past, still presents some uncertainties as regards both the analytical composition of the material and its chemical nature. It has therefore been suggested by Sir John Murray, to whom directly or indirectly all our knowledge of Red Clay is due, that the subject might with advantage be reopened from the chemical standpoint. The results of this revision, which was carried out in the Challenger Laboratory at Edinburgh, are set forth and discussed in the ensuing pages.

Let u be a real valued strong solution defined in a cylindrical domain of a linear second-order parabolic equation in two space variables with entire coefficients. Then it is shown that on compact subsets of its domain of definition u can be approximated arbitrarily closely in the maximum norm by an entire solution of the parabolic equation.

Mirror Gamow-Teller β-decay takes place from members of isospin multiplets to a common final state or to final states that are mirrors of each other. It is found that in these cases the intrinsic speed of the positon emission is systematically lower, by 10–15 per cent, than that of the mirror negaton emission. This difference could be due to a fundamental weak interaction effect, a second-class term in the weak hadronie current, or to a failure of exact symmetry in the nuclear structure. The influence of binding energy differences is studied in the latter context and it is concluded that they cannot be responsible for the empirical asymmetry. The effects of changes of configurational mix across the multiplets is also considered; very few estimates have ben explicitly made but it would be surprising if this effect turned out to be of constant sign and magnitude over the wide range of A involved. No present firm conclusion can be made but none of the empirical evidence is inconsistent with the reality of a largely momentum-transfer-independent second-class current.

A class of systems of two-dimensional singular integral equations with even kernels over bounded domains in the plane is studied. The applications include integral equations with the Bergman kernel function. The method of investigation is the following: the integral equation is reduced to a Riemann type boundary value problem for a first order elliptic system. This is solved by means of one-dimensional singular integral equations over the boundary curve. An adjoint problem is formulated, the Noetherian theorems are established, and a formula for the index is given.

The semi-linear equation Δu − λu + h(x)uσ = 0 is studied on all of d-dimensional Euclidean space. In the bifurcation problem a non-trivial solution is sought for small λ which tends to zero with λ. The asymptotic dependence of the solution on λ is examined. For fixed λ = 1 the existence of non-degenerate non-trivial solutions is proved for generic measurable h(x) sufficiently near to a constant, provided d = 1 or 3. The two problems are seen to be interdependent. The bifurcation problem at λ = 0 is particularly interesting as the linearised equation is of non-Fredholm type.

In October 1890, Dr Bramwell sent to the Laboratory a small quantity of urine, with a request that I should give him a report upon it. To him I am indebted for the opportunity of making the series of observations here recorded.

On examination, I found that the amount of proteid was enormous, and that it was made up almost entirely of a globulin :—

This condition of the urine persisted, with only slight changes, until the death of the patient.

In a note on “The Effect of Heat on an Infusible, Impalpable Powder,” by Professor P. G. Tait, in the Proceedings of the Royal Society of Edinburgh, vol. ix. p. 298, for the year 1876–77, Professor Tait points out that such a powder becomes very fluid under the action of heat, and behaves in many respects in the same way as a liquid would do—viz., convection currents are distinctly to be observed, and small particles of the powder are thrown up from the surface, in the same manner as we perceive little drops of water thrown up from the surface of a glass of soda-water. And Professor Tait then asks the question—If, supposing we had two such infusible, impalpable powders, would they diffuse into one another as do gases and liquids ? This is a question which as yet has not been answered. Professor Tait and I have been engaged in some experiments on the subject for some time, but the difficulties (chemical and physical) to be overcome are much greater than at first sight appear, and at present we are unable to say definitely whether they do so or not. But I think an answer may be obtained from another source. In some recent experiments I had occasion to have a number of Berlin porcelain crucibles and amorphous carbon in an impalpable powder kept in contact with each other at very high temperatures for from ten to twelve hours, with the following effect, that, although the crucibles did not become fused, but retained their exact original form, yet the carbon found its way to a considerable distance into the crucible, and some of the particles penetrated the crucible throughout. This was not a case, therefore, of fusion and mechanical mixture.

In this work we study the Cauchy problem in Gevrey spaces for a generalized class of equations that contains the case $b=0$ of the $b$-equation. For the generalized equation, we prove that it is locally well-posed for initial data in Gevrey spaces. Moreover, as we move to global well-posedness, we show that for a particular choice of the parameter in the equation the local solution is global analytic in both time and spatial variables.

The purpose of this paper is to describe various subspaces that are closely related to the absolutely continuous subspace of a Floquet operator. This paper generalises and extends several known results.