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.

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) 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 Peterson and are in the spirit of our results for the continuous analogue. (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.

Practical scheme for quantum computation with any two-qubit entangling gate

Which gates are universal for quantum computation? Although it is well known that certain gates on two-level quantum systems (qubits), such as the controlled-NOT, are universal when assisted by arbitrary one-qubit gates, it has only recently become clear precisely what class of two-qubit gates is universal in this sense. We present an elementary proof that any entangling two-qubit gate is universal for quantum computation, when assisted by one-qubit gates. A proof of this result for systems of arbitrary finite dimension has been provided by Brylinski and Brylinski; however, their proof relies on a long argument using advanced mathematics. In contrast, our proof provides a simple constructive procedure which is close to optimal and experimentally practical.

Practical and economic advantages of choice feeding systems for laying poultry

Poultry can be managed under different feeding systems, depending on the husbandry skills and the feed available. These systems include the following: (1) a complete dry feed offered as a mash ad libitum; (2) the same feed offered as pellets or crumbles ad libitum; (3) a complete feed with added whole grain; (4) a complete wet feed given once or twice a day; (5) a complete feed offered on a restricted basis; (6) choice feeding. Of all these, an interesting alternative to offering complete diets is choice feeding which can be applied on both a small or large commercial scale. Under choice feeding or free-choice feeding birds are usually offered a choice between three types of feedstuffs: (a) an energy source (e.g. maize, rice bran, sorghum or wheat); (b) a protein source (e.g. soyabean meal, meat meal, fish meal or coconut meal) plus vitamins and minerals and (c), in the case of laying hens, calcium in granular form (i.e. oyster-shell grit). This system differs from the modern commercial practice of offering a complete diet comprising energy and protein sources, ground and mixed together. Under the complete diet system, birds are mainly only able to exercise their appetite for energy. When the environmental temperature varies, the birds either over- or under-consume protein and calcium. The basic principle behind practising choice feeding with laying hens is that individual hens are able to select from the various feed ingredients on offer and compose their own diet, according to their actual needs and production capacity. A choice-feeding system is of particular importance to small poultry producers in developing countries, such as Indonesia, because it can substantially reduce the cost of feed. The system is flexible and can be constructed in such a way that the various needs of a flock of different breeds, including village chickens, under different climates can be met. The system also offers a more effective way to use home-produced grain, such as maize, and by-products, such as rice bran, in developing countries. Because oyster-shell grit is readily available in developing countries at lower cost than limestone, the use of cheaper oyster-shell grit can further benefit small-holders in these countries. These benefits apart, simpler equipment suffices when designing and building a feed mixer on the farm, and transport costs are lower. If whole (unground) grain is used, the intake of which is accompanied by increased efficiency of feed utilisation, the costs of grinding, mixing and many of the handling procedures associated with mash and pellet preparation are eliminated. The choice feedstuffs can all be offered in the current feed distribution systems, either by mixing the ingredients first or by using a bulk bin divided into three compartments.

The nonexistence of spurious solutions to discrete, two-point boundary value problems

We investigate difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations. We formulate conditions under which all solutions to the discrete problem satisfy certain a priori bounds which axe independent of the step-size. As a result, the nonexistence of spurious solutions are guaranteed. Some existence and convergence theorems for solutions to the discrete problem are also presented. (C) 2002 Elsevier Science Ltd. All rights reserved.

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.

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C: K--> 2(Y) a point-to-set mapping such that for any x is an element of K, C(x) is a pointed, closed, and convex cone in Y and int C(x) not equal 0. Given a mapping g : K --> K and a vector valued bifunction f : K x K - Y, we consider the implicit vector equilibrium problem (IVEP) of finding x* is an element of K such that f (g(x*), y) is not an element of - int C(x) for all y is an element of K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems. (C) 2003 Elsevier Science Ltd. All rights reserved.

On continuous selection problems for multivalued mappings with the local intersection property in hyperconvex metric spaces

In the paper we present two continuous selection theorems in hyperconvex metric spaces and apply these to study xed point and coincidence point problems as well as variational inequality problems in hyperconvex metric spaces.

A practical guide to exercise training for heart failure patients

Background: Exercise training has been shown to improve exercise capacity in patients with heart failure. We sought to examine the optimal strategy of exercise training for patients with heart failure. Methods: Review of the published data on the characteristics of the training program, with comparison of physiologic markers of exercise capacity in heart failure patients and healthy individuals and comparison of the change in these characteristics after all exercise training program. Results: Many factors, including the duration, supervision, and venue of exercise training; the volume of working muscle; the delivery mode (eg, continuous vs. intermittent exercise), training intensity; and the concurrent effects of medical treatments may influence the results of exercise training in heart failure. Starting in an individually prescribed and safely monitored hospital-based program, followed by progression to an ongoing and progressive home program of exercise appears to be the best solution to the barriers of anxiety, adherence, and ease of access encountered by the heart failure patient. Conclusions: Various exercise training programs have been shown to improve exercise capacity and symptom status in heart failure, but these improvements may only be preserved with an ongoing maintenance program.