The elliptic sine-Gordon equation in a half plane

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case. We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation. We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.

The Klein-Gordon equation in a domain with time-dependent boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time-dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.

The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.

Spectral analysis of the elliptic sine-Gordon equation in the quarter plane

We study the elliptic sine-Gordon equation in the quarter plane using a spectral transform approach. We determine the Riemann-Hilbert problem associated with well-posed boundary value problems in this domain and use it to derive a formal representation of the solution. Our analysis is based on a generalization of the usual inverse scattering transform recently introduced by Fokas for studying linear elliptic problems.

Productivity of crossbred dairy cows suckling their calves for 12 or 24 weeks post calving

An experiment on restricted suckling of crossbred dairy cows was conducted at the Livestock Research Centre, Tanga in northeast Tanzania. Thirty-six Bos taurus (Holstein Friesian and Jersey) x Bos indicus (East African Zebu) cows were allocated alternately as they calved to suckling their calves for either 12 or 24 weeks after calving. Cows grazed improved pastures and were offered 4 kg concentrate daily. Milking occurred twice daily by hand; calves were allowed to suck residual milk for 30 min following each milking. Calves were also allowed access to grazing and were offered a maximum of I kg concentrate daily to 24 weeks of age. Weaning age had no significant effect on lactation milk yield for human consumption, the mean (SE) yield being 1806 (102.0) L and 1705 (129. 1) L for 12- and 24-week weaning, respectively. Cows from the two treatments suffered similar losses of live weight and body condition score during lactation and neither group had returned to the original body condition score 40 weeks following calving. Post-partum anoestrous intervals were prolonged. Although not significant, cows suckling calves to 24 weeks had a mean interval to first oestrus extended by 38 days compared with cows suckling calves to 12 weeks. The mean (SE) daily live weight gains of the calves to 52 weeks were 263 (14.1) g/day and 230 (18.1) g/day for calves weaned at 12 and 24 weeks, respectively, such that 12-month weights were 119 (5.6) kg and 110 (7.3) kg, respectively. Twelve-week-weaned calves consumed more concentrate (p < 0.05) from 13 to 24 weeks than did 24-week weaned calves. Calculation of residual milk consumption removed by calves from birth to 12 weeks indicated that it accounted for 28% of total yield. No benefits in cow and calf performance and welfare were found to justify prolonging the suckling period to 24 weeks.

Longitudinal selenium status in healthy British adults: assessment using biochemical and molecular biomarkers

Human selenium (Se) requirements are currently based on biochemical markers of Se status. In rats, tissue glutathione peroxidase-1 (Gpx1) mRNA levels can be used effectively to determine Se requirements; blood Gpx1 mRNA levels decrease in Se-deficient rats, so molecular biology-based markers have potential for human nutrition assessment. To study the efficacy of molecular biology markers for assessing Se status in humans, we conducted a longitudinal study on 39 subjects (age 45 +/- 11) in Reading, UK. Diet diaries (5 day) and blood were obtained from each subject at 2, 8, 17 and 23 weeks, and plasma Se, glutathione peroxidase (Gpx3) enzyme activity, and selenoprotein mRNA levels were determined. There were no significant longitudinal effects on Se biomarkers. Se intake averaged 48 +/- 14 mu g/d. Plasma Se concentrations averaged 1.13 +/- 0.16 mu mol/l. Plasma Se v. energy-corrected Se intake (ng Se/kJ/d) was significantly correlated, but neither Gpx3 activity v. Se intake (ng Se/kJ/d) nor Gpx3 activity v. plasma Se was significantly correlated. Collectively, this indicates that subjects were on the plateaus of the response curves. Selenoprotein mRNAs were quantitated in total RNA isolated from whole blood, but mRNA levels for Gpx1, selenoprotein H, and selenoprotein W (all highly regulated by Se in rodents), as well selenoprotein P, Gpx3, and phospholipid hydroperoxide glutathione peroxidase were also not significantly correlated with plasma Se. Thus selenoprotein molecular biomarkers, as well as traditional biochemical markers, are unable to further distinguish differences in Se status in these Se replete subjects. The efficacy of molecular biomarkers to detect Se deficiency needs to be tested in Se-deficient populations.

Effects of enhanced consumption of fruit and vegetables on plasma antioxidant status and oxidative resistance of LDL in smokers supplemented with fish oil

Objective: To determine whether consumption of five portions of fruit and vegetables per day reduces the enhancement of oxidative stress induced by consumption of fish oil. Subjects: A total of 18 free-living healthy smoking volunteers, aged 18-63 y, were recruited by posters and e-mail in The University of Reading, and by leaflets in local shops. Design: A prospective study. Setting: Hugh Sinclair Unit of Human Nutrition, School of Food Biosciences, The University of Reading, Whiteknights PO Box 226, Reading RG6 6AP, UK. Intervention: All subjects consumed a daily supplement of 4 x 1 g fish oil capsules for 9 weeks. After 3 weeks, they consumed an additional five portions of fruits and vegetables per day, and then they returned to their normal diet for the last 3 weeks of the study. Fasting blood samples were taken at the ends of weeks 0, 3, 6 and 9. Results: The plasma concentrations of ascorbic acid, lutein, beta-cryptoxanthin, alpha-carotene and beta-carotene all significantly increased when fruit and vegetable intake was enhanced (P<0.05). Plasma concentrations of α-tocopherol, retinol and uric acid did not change significantly during the period of increased fruit and vegetable consumption. Plasma oxidative stability, assessed by the oxygen radical absorbance capacity (ORAC) assay, also increased from weeks 3-6 (P<0.001) but not in association with increases in measured antioxidants. Lag phase before oxidation of low-density lipoprotein (LDL) significantly decreased in the first 3 weeks of the study, reflecting the incorporation of EPA and DHA into LDL (P<0.0001). Subsequent enhanced fruit and vegetable consumption significantly reduced the susceptibility of LDL to oxidation (P<0.005). Conclusion: Fish oil reduced the oxidative stability of plasma and LDL, but the effects were partially offset by the increased consumption of fruit and vegetables.

Dietary supplementation with a natural carotenoid mixture decreases oxidative stress

Objective: To determine whether dietary supplementation with a natural carotenoid mixture counteracts the enhancement of oxidative stress induced by consumption of fish oil. Design: A randomised double-blind crossover dietary intervention. Setting: Hugh Sinclair Unit of Human Nutrition, School of Food Biosciences, The University of Reading, Whiteknights PO Box 226, Reading RG6 6AP, UK. Subjects and intervention: A total of 32 free-living healthy nonsmoking volunteers were recruited by posters and e-mails in The University of Reading. One volunteer withdrew during the study. The volunteers consumed a daily supplement comprising capsules containing fish oil (4 x 1 g) or fish oil (4 x 1 g) containing a natural carotenoid mixture (4 x 7.6 mg) for 3 weeks in a randomised crossover design separated by a 12 week washout phase. The carotenoid mixture provided a daily intake of beta-carotene (6.0 mg), alpha-carotene (1.4 mg), lycopene (4.5 mg), bixin (11.7 mg), lutein (4.4 mg) and paprika carotenoids (2.2 mg). Blood and urine samples were collected on days 0 and 21 of each dietary period. Results: The carotenoid mixture reduced the fall in ex vivo oxidative stability of low-density lipoprotein (LDL) induced by the fish oil (P = 0.045) and it reduced the extent of DNA damage assessed by the concentration of 8-hydroxy-2'-deoxyguanosine in urine (P = 0.005). There was no effect on the oxidative stability of plasma ex vivo assessed by the oxygen radical absorbance capacity test. beta- Carotene, alpha-carotene, lycopene and lutein were increased in the plasma of subjects consuming the carotenoid mixture. Plasma triglyceride levels were reduced significantly more than the reduction for the fish oil control (P = 0.035), but total cholesterol, HDL and LDL levels were not significantly changed by the consumption of the carotenoid mixture. Conclusions: Consumption of the natural carotenoid mixture lowered the increase in oxidative stress induced by the fish oil as assessed by ex vivo oxidative stability of LDL and DNA degradation product in urine. The carotenoid mixture also enhanced the plasma triglyceride-lowering effect of the fish oil.

Effects of chronic and acute fruit and vegetable juice consumption on cardiovascular disease risk factors

Diets low in fruit and vegetables are reportedly responsible for 2.7 million deaths annually from cardiovascular diseases (CVD) and certain cancers. A daily fruit and vegetable intake of five 80 g portions is recommended for chronic disease prevention. However, in the UK, average adult consumption is less than three portions. It is suggested that fruit juice should only count as one portion. However, fruit juices are a beneficial source of phytochemicals. The preliminary results of two randomized, controlled, crossover, dietary intervention studies investigating the effects of chronic and acute consumption of fruit and vegetable puree and juice based drinks (FVPJ) on bioavailability, antioxidant status, vascular reactivity, and risk factors for CVD are reported. In the first study, 39 volunteers consumed 200 ml FVPJ, or fruit-flavoured control, daily for six weeks. In the second study, 24 volunteers consumed 400 mL FVPJ, or sugar-matched control, on the morning of the study day. Blood and urine samples were collected throughout both studies and real-time measurements of vascular tone were performed using laser Doppler imaging with iontophoresis. Overall, the studies showed that the fruit and vegetable puree and juice based drink increased dietary phytochemicals. There was a trend towards increased vasodilation following both acute and chronic fruit juice consumption. Measurements of antioxidant status, oxidative stress and other cardiovascular disease risk factors are currently being determined.

The Dirichlet-to-Neumann map for the elliptic sine Gordon

We analyse the Dirichlet problem for the elliptic sine Gordon equation in the upper half plane. We express the solution $q(x,y)$ in terms of a Riemann-Hilbert problem whose jump matrix is uniquely defined by a certain function $b(\la)$, $\la\in\R$, explicitly expressed in terms of the given Dirichlet data $g_0(x)=q(x,0)$ and the unknown Neumann boundary value $g_1(x)=q_y(x,0)$, where $g_0(x)$ and $g_1(x)$ are related via the global relation $\{b(\la)=0$, $\la\geq 0\}$. Furthermore, we show that the latter relation can be used to characterise the Dirichlet to Neumann map, i.e. to express $g_1(x)$ in terms of $g_0(x)$. It appears that this provides the first case that such a map is explicitly characterised for a nonlinear integrable {\em elliptic} PDE, as opposed to an {\em evolution} PDE.