Existence of Multiple Solutions for Second Order Boundary Value Problems

We give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y" + f(x, y, y') = 0, x epsilon [0, 1], y(0) = 0 = y(1). In the special case f(x, y, y') = f(y) greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and of Lakshmikantham et al.

On the minimality of the p-harmonic map - x/vertical bar x vertical bar : B-n -> Sn-1

We prove that for any real number p with 1 p less than or equal to n - 1, the map x/\x\ : B-n --> Sn-1 is the unique minimizer of the p-energy functional integral(Bn) \delu\(p) dx among all maps in W-1,W-p (B-n, Sn-1) with boundary value x on phiB(n).

Effects of alkalis, oxidants and urea on the nutritive value of rhodes grass (Chloris gayana cv. Callide)

Two in sacco experiments were conducted to evaluate the impact on the nutritive value of rhodes grass hay (Chloris gayana cv. Callide) of treatment with alkalis or oxidants. In Experiment 1, three alkalis (Ca(OH)(2), NaOH, CaO) and two oxidants (NaOCl and H2O2) were applied at levels of 0, 20, 40, 60 or 80 g/kg of dry matter (DM). NaOH, Ca(OH)(2) and CaO had negative linear effects (P < 0.05) on the neutral detergent fibre (NDF) content and positive linear effects (P < 0.05) on the 48 h in sacco disappearances of DM, organic matter (OM), NDF and acid detergent fibre (ADF). NaOCl reduced (P < 0.05) NDF content but had no effect (P > 0.05) on the in sacco disappearances. H2O2 had no effect (P > 0.05) on the composition or digestibility of rhodes grass hay. In Experiment 2, effects of urea (0, 20, 40, 60 and 80 g urea/kg DM) and water (250, 500 and 750 g/kg DM) treatment of rhodes grass hay were examined in a 5 x 3 factorial experiment. Significant interactions between water and urea (P < 0.05) occurred for concentrations of crude protein (CP) and NDF, and 48 h in sacco disappearances of DM, OM (OMD) and NDE The combinations of water (g/kg DM) and urea (g/kg DM) that resulted in the highest concentrations of CP (281 g/kg DM) and OMD (747 g/kg DM) were 250 + 80 and 500 + 80, respectively. NaOH, Ca(OH)(2), CaO and urea significantly alter the NDF content and digestibility of rhodes grass hay, and urea also increases its CP content. Overall, NaOH was the most efficacious, followed by Ca(OH)(2), CaO, urea, NaOCl and H2O2. Crown Copyright (C) 2002 Published by Elsevier Science B.V. All rights reserved.

How well does B-type natriuretic peptide predict death and cardiac events in patients with heart failure: systematic review

Objective To assess how well B-type natriuretic peptide (BNP) predicts prognosis in patients with heart failure. Design Systematic review of studies assessing BNP for prognosis m patients with heart failure or asymptomatic patients. Data sources Electronic searches of Medline and Embase from January 1994 to March 2004 and reference lists of included studies. Study selection and data extraction We included all studies that estimated the relation between BNP measurement and the risk of death, cardiac death, sudden death, or cardiovascular event in patients with heart failure or asymptomatic patients, including initial values and changes in values in response to treatment. Multivariable models that included both BNP and left ventricular ejection fraction as predictors were used to compare the prognostic value of each variable. Two reviewers independently selected studies and extracted data. Data synthesis 19 studies used BNP to estimate the relative risk of death or cardiovascular events in heart failure patients and five studies in asymptomatic patients. In heart failure patients, each 100 pg/ml increase was associated with a 35% increase in the relative risk of death. BNP was used in 35 multivariable models of prognosis. In nine of the models, it was the only variable to reach significance-that is, other variables contained no prognostic information beyond that of BNP. Even allowing for the scale of the variables, it seems to be a strong indicator of risk. Conclusion Although systematic reviews of prognostic studies have inherent difficulties, including die possibility of publication bias, the results of the studies in this review show that BNP is a strong prognostic indicator for both asymptomatic patients mid for patients with heart failure at all stages of disease.

Three-point boundary value problems for second-order, ordinary, differential equations

We establish existence results for solutions to three-point boundary value problems for nonlinear, second-order, ordinary differential equations with nonlinear boundary conditions. (C) 2001 Elsevier Science Ltd. All rights reserved.

Relationships between crude protein and mineral concentrations in alfalfa and value of alfalfa silage as a mineral source for periparturient cows

One hundred and eight samples from three cultivars of alfalfa were obtained from three cuttings in 1996-1998 to evaluate the relationship between crude protein (CP) and mineral concentrations of alfalfa with cutting and maturation. The CP content drastically decreased from 27.9 to 11.4% on DM with maturity. Highly positive correlations were observed between CP and K in the first and the second cutting of alfalfa. The Ca content remained almost constant throughout the growth period. Four multiparous Holstein cows were assigned an alfalfa silage diet or an orchardgrass silage diet from 3 weeks prepartum to 1 week postpartum to examine the effect on the mineral balance. In the prepartum and postpartum diet, the roughage to concentrate ratio was 70:30 and 50:50, with alfalfa being 50 and 100% of the roughage, respectively. The alfalfa contained 1.93% of K. No metabolic disorders occurred, but the body weight decreased drastically from 1 to 6 days postpartum with each diet because of the high milk production immediately after the parturition. Positive retention of N, Ca, P, Mg, and K was observed prepartum, whereas the cows had negative N and mineral retention from 2 to 4 days postpartum. The Ca and P absorption, and Mg retention of cows with the alfalfa diet were higher than with the grass diet. The plasma Ca and inorganic P were not affected by diet, but the plasma PTH at parturition and plasma hydroxyproline from 1 week prepartum to 1 week postpartum were higher with the alfalfa diet. These results suggest that the low K alfalfa is suitable not only to prevent the incidence of milk fever but also to increase Ca, P and Mg utilization of periparturient cows, but the mineral supplementation is needed for the postpartum cows immediately after the parturition. (C) 2001 Elsevier Science B.V. All rights reserved.

The nutritive value of Rhodes grass (Chloris gayana) when treated with CaO, NaOH or a microbial inoculant and offered to dairy heifers as big-bale silage

A series of laboratory and animal studies examined the use of chemical and biological agents to enhance the digestibility of Rhodes grass (grass) cut at 60 (young) and 100 (mature) days of regrowth and ensiled as big round bales. The treatments included an untreated control (C), a microbial inoculant (I), NaOH, CaO and NaOH plus inoculant (NaOH + I). Inoculant was grown anaerobically, using a starter culture of rumen fluid from cattle given Rhodes grass. Treatments C, 1, NaOH, NaOH + I, were offered separately to twelve dairy heifers, in a 3 X 4 randomized complete block design, repeated twice for each grass silage. C and I had substantial mould growth, compared with no visible mould in NaOH or NaOH + 1. CaO treatment was effective in preventing mould growth, but had little effect on the chemical composition and in sacco digestibility of mature grass silage. NaOH reduced NDF content and increased in sacco digestibility (P < 0.05) but not the in vivo digestibility (P > 0.05) of both mature- and young-grass silage. The effects of other treatments on nutritive value were non-significant at both stages of maturity. NaOH increased the intake of mature-grass silage by 24-26% (P < 0.05), but had little effect on the intake of young-grass silage (P > 0.05). Treatment I consistently reduced grass silage intake (P < 005) for young-grass silage. The findings of these studies show that treating mature Rhodes grass with NaOH will improve its nutritive value and reduce mould growth in conserved herbage. However none of the treatments in this study had any consistently positive effects on the in vivo nutritive value or storage quality of young-grass silage.

Reconciling value estimates from the discounted cash flow model and the residual income model

This paper examines why practitioners and researchers get different estimates of equity value when they use a discounted cash flow (CF) model versus a residual income (RI) model. Both models are derived from the same underlying assumption -- that price is the present value of expected future net dividends discounted at the cost of equity capital -- but in practice and in research they frequently yield different estimates. We argue that the research literature devoted to comparing the accuracy of these two models is misguided; properly implemented, both models yield identical valuations for all firms in all years. We identify how prior research has applied inconsistent assumptions to the two models and show how these seemingly small errors cause surprisingly large differences in the value estimates. [ABSTRACT FROM AUTHOR]

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

Existence of multiple solutions for finite difference approximations to second-order boundary value problems

Error condition detected We consider discrete two-point boundary value problems of the form D-2 y(k+1) = f (kh, y(k), D y(k)), for k = 1,...,n - 1, (0,0) = G((y(0),y(n));(Dy-1,Dy-n)), where Dy-k = (y(k) - Yk-I)/h and h = 1/n. This arises as a finite difference approximation to y" = f(x,y,y'), x is an element of [0,1], (0,0) = G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g(0), g(1)) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0. (C) 2003 Elsevier Science Ltd. All rights reserved.

The value of screening in siblings of patients with abdominal aortic aneurysm

Objectives. This study aimed to determine the incidence of abdominal aortic aneurysm (AAA) in a large group of siblings of Australian AAA patients to determine if screening in this group is justified. Methods. 1254 siblings of 400 index AAA patients were identified and offered aortic ultrasound screening. An age and sex matched control group was recruited from patients having abdominal CT scans for non-vascular indications. AAA was defined by an infrarenal aortic diameter of greater than or equal to3 cm or a ratio of the infrarenal to suprarenal aortic diameter of greater than or equal to2.0. A ratio of 1.0-1.5 was considered normal, and a ratio of >1.5 to