In this paper we study some very weak notions of differentiability arising in connection with the spatial regularity of flows associated with non-smooth vector fields. The main difference from other similar concepts, also studied in a Sobolev setting, is that the convergence of difference quotients has to be understood as convergence in measure. We show in particular that the classical approximate differentiability is a property strictly stronger than approximate differentiability in measure.

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.

When an incident Herglotz wave function scatters from a periodic Lipschitz continuous surface with a Dirichlet boundary condition, the classical (quasi-)periodic solution theory for scattering from periodic structures does not apply, since the incident field lacks periodicity. Relying on the Bloch transform, we provide a solution theory in H1 for this scattering problem. First, we prove conditions guaranteeing that incident Herglotz wave functions propagating towards the periodic structure have traces in H1/2 on the periodic surface. Second, we show that the solution to the scattering problem can be decomposed by the Bloch transform into periodic components that solve a periodic scattering problem. Third, these periodic solutions yield an equivalent characterization of the solution to the original non-periodic scattering problem, which allows, for instance, new characterizations of the Rayleigh coefficients of each of the periodic components to be shown. A corollary of our results is that under the conditions mentioned above the operator that maps densities to the restriction of their Herglotz wave function on the periodic surface is always injective; this result generally fails for bounded surfaces.

Under some natural hypotheses, we show that if the (Nemitsky-) operator associated with an elliptic system is pseudomonotone, then the system has to be quasimonotone. Conversely, if the system satisfies a strict quasimonotonicity condition, then an existence proof of K.-W. Zhang contains the arguments to verify the pseudomonotonicity of the operator. We present a simplified proof of this fact under more general assumptions.

The paper discusses the asymptotic behaviour of solutions of the functional differential equation

where a is a complex constant, 0<λ<1, and b is a constant such that Re b = 0, but b ≠ 0.

The Hilbert boundary value problem for a first order nonlinear elliptic system in the plane with linear boundary conditions of nonnegative index is (under suitable side conditions uniquely) solved by use of the Newton imbedding method. This constructive method is based on an a priori estimate which arises from an integral representation formula for C1-functions first developed by Haack and Wendland. The approximation procedure yields an error estimate too.

In this paper we study the problem

where Ω = ℝ N or Ω = B 1, N ⩾ 3, p > 1 and . Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.

By using a general result on the monotonicity of quotients of power series, our aim is to prove some monotonicity and convexity results for the modified Struve functions. Moreover, as consequences of the above-mentioned results, we present some functional inequalities as well as lower and upper bounds for modified Struve functions. Our main results complement and improve the 1998 results of Joshi and Nalwaya.

The Ambarzumyan theorem connecting the Sturm–Liouville problem and the corresponding problem associated with the Fourier differential equation is extended to a class of second order matrix differential systems.

We discuss the spectral properties of a class of sequences of what we call ‘spectral’ type. We introduce an effective method to calculate the zeta invariants for this type of sequence. Such invariants are given in terms of some new and old special functions, and we consider a number of examples in which we study the properties of these special functions.

We give examples of symplectic manifolds which are also non-trivial principal torus-bundles with Lagrangian fibres. These bundles are examples of spaces with an obstruction to the global existence of action-angle variables.

In June 1904 I received an invitation from Sir John Murray to visit Scotland and spend three or four weeks in exploring the Scottish lakes, in order to make a comparison between them and the Danish lakes : he was of opinion that such a comparison of the lakes of a highland and a lowland country, which had hitherto not been attempted, would lead to some interesting results. The admirable bathymetrical and physical explorations carried on by Sir John Murray in Scotland, and more especially in Loch Ness, being far advanced, the question as to the scope of the biological observations called for consideration ; so he desired me to indicate, from the impressions derived during my visit, my views as to the most useful lines of investigation that might be taken up with reference to the biology of the Scottish lakes. I was much interested in the task imposed upon me, and at the same time gratified at the prospect of assisting in the design of the biological explorations in the lakes of a foreign country ; and as it was of the greatest significance to me to learn the nature of alpine lakes, I immediately accepted the invitation. I spent three weeks in Scotland,—the first two at, Fort Augustus, on Loch Ness, and the third in Edinburgh. From Fort Augustus I explored the lakes of the Caledonian Canal, and thus became acquainted with alpine lakes; from Edinburgh I explored a few lowland lakes, especially Loch Leven. The steamer Mermaid, belonging to the Marine Biological Station at Millport, fully equipped for deep-sea work, under the direction of Dr Gemmill, was sent into the Caledonian Canal, and many hauls were taken with the dredge and trawl, as well as with different kinds of tow-nets, in Lochs Lochy, Oich, and Ness, down to the greatest depths (500 and 750 feet).

I much regret it has not been possible for me, during this summer, experimentally to determine the best forms and sizes of the details in the construction of the C screen; nor have I been able to keep a continuous record of its readings. Only a few observations have been made at intervals with it, as originally constructed, and shown in fig. 1 of this paper, and with the Stevenson screens. The result of all these trials is to confirm the conclusions already arrived at. The C screen always gave the lowest readings when there was any radiation.

The cellular processes of ovulation, maturation and fertilisation of the ovum in the albino mouse have been the subject of repeated and exact studies, and are now sufficiently well known. The biological phenomena of propagation in the mouse, i.e. the sexual cycle, œstrus, and copulation, have likewise been described in detail, but our knowledge concerning the period of gestation and the moment of parturition is still very inadequate.

In a previous paper on the Cardinal Function it was proved that, calling

the cardinal function of the table of values αr at the point α + rw], “the cardinal function of the table of values [C(rw′) at the point rw′] is a function C1(x) which coincides with C(x) provided that 0<w′<w, and Σ | αn/n |, Σ |α−n/n| are convergent.”

We consider a quasilinear elliptic problem of the form

where λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

The radial distribution of the atoms (or molecules) of liquids, as of solids, can be found from X-ray scattering photographs (Zernike and Prins, 1927; Warren and Gingrich, 1934), and in this way many such distributions have been determined experimentally (Harvey, 1938, 1939; Gingrich, 1940; references in Coulson and Rushbrooke, 1939). The results are generally given as distribution curves showing p(r) or r2p(r) as a function of r, where 4πr2p(r)dr measures the probability that two arbitrarily selected atoms (or molecules) of the liquid are distant r to r + dr apart. Such distribution curves show a sequence of peaks—for ρ(r) these are usually of diminishing amplitude—and for atomic liquids successive peaks may conveniently be ascribed to successive co-ordination shells (of atoms) about any (arbitrarily selected) central atom. The position and precise shape of the peaks depends, of course, upon the temperature as well as the substance of the liquid.

We investigate different types of periodic solutions of a fourth-order, nonlinear differential equation, which has been proposed as a model for travelling waves in suspension bridges. We develop a shooting argument, which enables us to prove the existence of two families of multibump periodic solutions, each containing a countably infinite number of different solutions

We study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

In this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem −Δu + u = up + ƒ(z) in Ω has at least two positive solutions, where

a bounded C1,1 domain S = × ω Rn, D is a bounded C1,1 domain in Rm+n such that D ⊂⊂ S and Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing that D is small.