We characterize, in the context of rearrangement invariant spaces, the optimal range space for a class of monotone operators related to the Hardy operator. The connection between the optimal range and the optimal domain for these operators is carefully analysed.

We give an L2 x L2 → L2 convolution estimate for singular measures supported on transversal hypersurfaces in ℝn, which improves earlier results of Bejenaru et al. as well as Bejenaru and Herr. The quantities arising are relevant to the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp–Lieb inequalities with explicit constants in the same spirit.

We consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points, specified by a finite incidence matrix E. Similarly the allowable discontinuities of the piecewise polynomials are specified by a finite incidence matrix F. We first find necessary conditions on (E, F) for the problem to be poised, that is to have a unique solution for any given data. The main result gives sufficient conditions on (E, F) for the problem to be poised, generalising a well-known result of Atkinson and Sharma. To this end we prove some results involving estimates of the numbers of zeros of the relevant piecewise polynomials.

In this note semi-bounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second-order partial differential equations on domains with non-compact boundaries.

An idempotent-separating congruence μ is studied further in this paper. It is shown to satisfy special properties with respect to regular elements and to group-bound elements. It is shown that for any semigroup S, μ is the identity congruence on S/μ. From this, it can be shown that S/μ is fundamental for any semigroup S. Some alternative characterizations of μ are given and applied to yield sufficient conditions for a subsemigroup T of S to satisfy μ (T) = μ (S) ∩ (T × T), whence T is fundamental if S is fundamental.

We show the $L^1$ contraction and comparison principle for weak (and, more generally, renormalized) solutions of the elliptic–parabolic problem $j(v)_t-\text{div}(\nabla w+F(w))=f(t,x)$, $w=\varphi(v)$ in $(0,T)\times\varOmega\subset \mathbb{R}^+\times\mathbb{R}^N$ with inhomogeneous Dirichlet boundary datum $g\in L^2(0,T;W^{1,2}(\varOmega))$ for $w$ (the boundary datum is taken in the sense $w-g\in L^2(0,T;H^{1}_0(\varOmega))$) and initial datum $j_o\in L^1(\varOmega)$ for $j(v)$. Here $\varphi$ and $j$ are non-decreasing, and we assume that $F$ is just continuous.

Our proof consists in doubling of variables in the interior of $\varOmega$ as introduced by Carrillo in 1999, and in a careful treatment of the flux term near the boundary of $\varOmega$. For the latter argument, the result is restricted to the linear dependence on $\nabla w$ of the diffusion term. The proof allows for a wide class of domains $\varOmega$, including, for example, weakly Lipschitz domains with Lipschitz cracks.

We obtain the corresponding results for the associated stationary problem and discuss on generalization of our technique to the case of nonlinear diffusion operators.

For a symmetric linear relation S with a directing mapping, the notion of a spectral function is defined by means of a Bessel–Parseval inequality, and a description of all such spectral functions is given. As an application, we describe the set of all spectral functions of a canonical regular first order differential system.

It is shown that a cantilevered beam with weak viscoelastic damping of Boltzmann-type can be uniformly stabilised by velocity feedback applied as a shearing force at the free end of the beam. Estimates for the viscoelastic energy are derived using the energy multiplier method. The energy decay is related to the decay of the relaxation modulus associated with the viscoelastic material.

We study an abstract mixed variational problem, the set of the Lagrange multipliers being dependent on the solution. The problem consists of a system of a variational equation and a variational inequality. We prove the existence of the solution based on a fixed-point technique for weakly sequentially continuous maps. We then apply the abstract result to the weak solvability of a boundary-value problem that models the frictional contact between a cylindrical deformable body and a rigid foundation.

§ 1. In the practical calculation of the periods and nodes of the lakes we have hitherto examined it has been found that the use of the Seiche Functions gives the best results. Indeed, as will appear from details presently to be submitted to the Society by Mr E. M. Wedderburn and myself, the agreement between theory and observation, so far as we have gone, is beyond what might reasonably have been expected. Also the phenomena of concave lakes, i.e. such as have no shallows or points of minimum depth, are easily deducible from the formulæ given in an abstract (Proc. R.S.E., vol. xxv. p. 328, 6th Oct. 1904) which I communicated to the Society on 18th July 1904. On the other hand, the theory of convex lakes is less easy of manipulation, chiefly owing to the difficulty in calculating the roots of the equations

1. The following note may be of some interest as helping in the elucidation of the rather complex analytical questions involved in the derivation of the modified Lagrangian function for a dynamical system. The results derived also have some bearing on the various questions involved in the transformation theory based on Hamilton's equations of motion. The discussion is given for the simplest type of system, but it can be easily generalised to the less restricted cases covered by the results.

The land nemerteans of the world comprise only twelve species and at present all are referred to the genus Geonemertes Semper. Most of the species occur in the Australia, New Zealand, and Malaysian regions, with a single species in Bermuda. European occurrences are rare and apparently due to introductions by man. G. chalicophora Graff, for example, is unknown outside glass-houses in Germany, Austria, Czechoslovakia (Stammer, 1934), and the Botanic Gardens, Dublin, Ireland (Southern, 1911). One other species, G. dendyi Dakin, which was originally described from a single female discovered in “a valley in the Darling Range not far from Perth,” West Australia (Dakin, 1915), was later found in glass-houses in the Botanic Gardens at Breslau, Germany (Stammer, 1934). More recently this worm has been found by us to be breeding in the open in wild situations around Swansea, Wales. We believe this occurrence to be the first record of a land nemertean established in the temperate climate of Europe.

In meteorological observations the temperature of the air is of the first importance, and it is a subject to which a great deal of attention is given, thousands of temperature observations being made and recorded every day. It is therefore desirable that these observations shall be as correct as possible. At first sight nothing seems more simple than to take the temperature of the air. All that appears necessary is to hang up a correct thermometer of any construction, anywhere out of the sun, and the thermometer will then indicate the temperature of the air where the instrument is placed. We shall presently see that this is very far from being the case; and not only so, but we shall find that it is difficult to get two thermometers which will give the same readings, when hung near each other in the open air, even though they agree perfectly with each other when placed in water.

The symmetry set of a plane curve y is the locus of centres of circles which either (i) are tangent to y at at least two distinct points, or (ii) have at least 4-point (A3) contact with y somewhere. The points (ii) also lie at cusps of the evolute of γ and the symmetry set, together with the evolute, can be studied by regarding their union as a full bifurcation set. For information on symmetry sets, see [2, 4, 5, 7, 8].

Here we shall use a theorem of Ozawa [9] to deduce global formulae relating three features of symmetry sets. The transitions which occur on the symmetry set of a generic one-parameter family of plane curves have been found [4], and more insight into the global formulae can be gained by tracing them through the transitions and verifying that they are invariant (see Section 2). We also briefly introduce symmetry sets of plane polygons in Section 3, and show that, contrary to first impressions, the same global formulae hold there.

We study certain perturbations of the differential equation Δu − u + up = 0 on all of n-dimensional Euclidean space. Conditions are obtained which ensure the existence of a solution to the perturbed equation near a given solution to the unperturbed equation. We have to overcome degeneracy of the unperturbed solution and lack of smooth dependence on the perturbation parameter. An abstract version of the argument is sketched in a functional-analytic setting related toequivariant bifurcation theory. We consider also a smooth perturbation with several parameters and study the singularities of the mapping which maps each solution to its associated parameters.

Let f be a transcendental meromorphic function on the complex plane ℂ, let a be a non-zero finite complex number and let n and k be two positive integers. In this paper, we prove that if n≥k+1, then assumes each value b∈ℂ infinitely often. Also, the related normal criterion for families of meromorphic functions is given. Our results generalize the related results of Fang and Zalcman.

Global weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

§ 1. The scientific world is practically unanimous in believing that all tangible or palpable matter, molar matter as we may call it, consists of groups of mutually interacting atoms or molecules. This molecular constitution of matter is essentially a deviation from homogeneousness of substance, and apparent homogeneousness of molar matter can only be homogeneousness in the aggregate.

We prove the existence of multiple bound states of the nonlinear Schrödinger equation −Δu + V(x)u = f(u). Here the linear potential V is continuous and bounded from below, and the nonlinearity f is of asymptotically linear type. We show that, under certain assumptions on the spectrum of the Schrödinger operator −Δ + V and the asymptotic behaviour of f(u)/u, the above equation has at least four non-trivial solutions, two of them sign changing.

We consider a class of convex non-linear boundary value problems of the form

where L is a linear, uniformly elliptic, self-adjoint differential expression, f is a given non-linear function, B is a boundary differential expression of either Dirichlet or Neumann type and D is a bounded open domain with boundary ∂D. Particular problems of this class arise in the process of thermal combustion [8].

In this paper we show that stable solutions of this class can be bounded from below (above) by a monotonically increasing (decreasing) sequence of Newton (Picard) iterates. The possibility of using these schemes to construct unstable solutions is also considered.