The snow surface roughness at centimetre and millimetre scales is an important parameter related to wind transport, snowdrifts, snowfall, snowmelt and snow grain size. Knowledge of the snow surface roughness is also of high interest for analyzing the signal from radar sensors such as SAR, altimeters and scatterometers. Unfortunately, this parameter has seldom been measured over snow surfaces. The techniques used to measure the roughness of other surfaces, such as agricultural or sand soils, are difficult to implement in polar regions because of the harsh climatic conditions. In this paper we develop a device based on a laser profiler coupled with a GPS receiver on board a snowmobile. This instrumentation was tested successfully in midre Lovénbreen, Svalbard, in April 2006. It allowed us to generate profiles of 3 km sections of the snow-covered glacier surface. Because of the motion of the snowmobile, the roughness signal is mixed with the snowmobile signal. We use a distance/frequency analysis (the empirical mode decomposition) to filter the signal. This method allows us to recover the snow surface structures of wavelengths between 4 and 50 cm with amplitudes of >1 mm. Finally, the roughness parameters of snow surfaces are retrieved. The snow surface roughness is found to be dependent on the scales of the observations. The retrieved RMS of the height distribution is found to vary between 0.5 and 9.2 mm, and the correlation length is found to be between 0.6 and 46 cm. This range of measurements is particularly well adapted to the analysis of GHz radar response on snow surfaces.

Recovery Ice Stream has multiple branches reaching far into the East Antarctic ice sheet. We use new airborne and ground-based geophysics to give the first comprehensive overview of the upper catchment and, by constraining the physical setting, to advance our understanding of the controlling mechanisms for the onset of fast flow. The 400 km wide ice stream extends towards the Recovery Subglacial Lakes, a region characterized by a crustal boundary, a change in bed roughness, a bedrock topographic step and four topographic basins (A–D), three of which (A–C) contain subglacial water. All these characteristics are considered potential causal mechanisms that contribute to the onset of fast flow. In Lakes B and C the subglacial water is located in basins with sharp downstream ridges, in contrast to the gently sloping ridge on the downstream margin of Lake A. The fastest-flowing branch of the ice stream emanates from Lake A. The presence of multiple causal mechanisms along the four Recovery Lakes allows us to identify basal water as a dominant factor for the onset of fast flow, but only if it is stored in a shallow-sided basin where it can lubricate the flow downstream. Relatively minor topographic barriers appear to inhibit streaming.

Periods of rapid growth seen during the early stages of fetal development, including cell proliferation and differentiation, are greatly influenced by the maternal environment. We demonstrate here that over-nutrition, specifically exposure to a high-fat diet in utero, programed the extent of atherosclerosis in the offspring of ApoE*3 Leiden transgenic mice. Pregnant ApoE*3 Leiden mice were fed either a control chow diet (2.8% fat, n=12) or a high-fat, moderate-cholesterol diet (MHF, 19.4% fat, n=12). Dams were fed the chow diet during the suckling period. At 28 days postnatal age wild type and ApoE*3 Leiden offspring from chow or MHF-fed mothers were fed either a control chow diet (n=37) or a diet rich in cocoa butter (15%) and cholesterol (0.25%), for 14 weeks to induce atherosclerosis (n=36). Offspring from MHF-fed mothers had 1.9-fold larger atherosclerotic lesions (P<0.001). There was no direct effect of prenatal diet on plasma triglycerides or cholesterol; however, transgenic ApoE*3 Leiden offspring displayed raised cholesterol when on an atherogenic diet compared with wild-type controls (P=0.031). Lesion size was correlated with plasma lipid parameters after adjustment for genotype, maternal diet and postnatal diet (R 2=0.563, P<0.001). ApoE*3 Leiden mothers fed a MHF diet developed hypercholesterolemia (plasma cholesterol two-fold higher than in chow-fed mothers, P=0.011). The data strongly suggest that maternal hypercholesterolemia programs later susceptibility to atherosclerosis. This is consistent with previous observations in humans and animal models.

The main results of the paper determine all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles and such that certain combinations of derivatives of f have few non-real zeros.

Let f be a real meromorphic function of infinite order in the plane, with finitely many zeros and non-real poles. Then f″ has infinitely many non-real zeros.

It is shown that if f is an analytic function of sufficiently small exponential type in the right half-plane, which takes integer values on a subset of the positive integers having positive lower density, then f is a polynomial.

We prove that for a function f(z) transcendental and meromorphic in the plane and not of the form exp(az + b), we have either N(r, 1/ff″)≠0(T(r, f′/f)) or .

For any sequence (aj) of complex numbers and for any ρ > ½, we construct an entire function F with the following properties. F has order ρ, mean type, each aj is a deficient value of F, and F is given by F(z)=f(g(z)), where f and g are transcendental entire functions. This complements a result of Goldstein. We also construct, for any ρ>½, an entire function G of order p, mean type, such that liminf,→ ∞ T(r, G)/T(r, G′)>1.

Let $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either

1. $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or

2. $f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.

AMS 2000 Mathematics subject classification: Primary 30D35

Our starting point is the differential equation

where A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],

for all , b∈C\{0}, at least outside a set of r of finite measure.

We determine all functions f(z) meromorphic in the plane such that f′(z)/f(z) has finite order and f(z) and F(z) have only finitely many zeros, where F(z) = f″(z) + Af(z) for some constant A.

Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = Δf(z) = f(z + 1) − f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f. The results may be viewed as discrete analogues of existing theorems on the zeros of f' and f'/f.

Let f be transcendental and meromorphic in the plane and let the non-homogeneous linear differential polynomials F and G be defined bywhere k,n ∈ N and a, b and the aj, bj are rational functions. Under the assumption that F and G have few zeros, it is shown that either F and G reduce to homogeneous linear differential polynomials in f + c, where c is a rational function that may be computed explicitly, or f has a representation as a rational function in solutions of certain associated linear differential equations, which again may be determined explicitly from the aj, bj and a and b.

A result is proved which implies the following conjecture of Osgood and Yang from 1976: if $f$ and $g$ are non-constant entire functions, such that $ T(r, f) = O(T(r, g))$ as $ r \to \infty $ and such that $g(z) \in {\mathbb Z}$ implies that $f(z) \in {\mathbb Z}$, then there exists a polynomial $G$ with coefficients in ${\mathbb Q}$, such that $G({\mathbb Z}) \subseteq {\mathbb Z}$ and $f = G \circ g$.