On a class of boundary value problems involving the p-biharmonic operator

A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p>1. It describes various problems in the theory of elasticity, e.g., the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p−1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular boundary conditions which are frequently considered in the literature.     

Lower bounds on the worst-case complexity of some oracle algorithms

To give a proper definition of the complexity of very general computational problems such as optimization problems over arbitrary independence systems or fixed-point problems for continuous functions, it is useful to consider the input for these problems as “oracles” R which can be called by the algorithms for some values x ∈ X and which then give back some information R(x) about x, e.g. whether x belongs to the independence system or the point into which x is mapped by the continuous function. A lower bound on the complexity of an algorithm using an oracle R is the number of calls on R in the worst case. Using this technique it is shown that there is no polynomial approximative algorithm for the maximization problem over a general independence system which has a better worst-case behaviour than the greedy algorithm. Moreover several formalizations of the problem of approximating a fixed point of a continuous function are considered, and it is shown that none of these problems can be solved by a bounded algorithm.     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Optimization problems for general simple population with n-impulsive harvest

In this paper, n-impulsive harvest problems of general simple population are discussed by models with Dirac function. The optimal impulsive harvest policies to protect the renewable resource better are obtained under conditions of fixed quantity per impulsive harvest. Then, a concept of the sequence for ?-optimal harvest moments for general simple population is presented which is beneficial to protect resource better and sustainable development. Finally, we apply the conclusions to some special models.     

Better bases for radial basis function interpolation problems

Radial basis function interpolation involves two stages. The first is fitting, solving a linear system corresponding to the interpolation conditions. The second is evaluation. The systems occurring in fitting problems are often very ill-conditioned. Changing the basis in which the radial basis function space is expressed can greatly improve the conditioning of these systems resulting in improved accuracy, and in the case of iterative methods, improved speed, of solution. The change of basis can also improve the accuracy of evaluation by reducing loss of significance errors. In this paper new bases for the relevant space of approximants, and associated preconditioning schemes are developed which are based on Floater’s mean value coordinates. Positivity results and scale independence results are shown for schemes of a general type. Numerical results show that the given preconditioning scheme usually improves conditioning of polyharmonic spline and multiquadric interpolation problems in R² and R³ by several orders of magnitude. The theory indicates that using the new basis elements (evaluated indirectly) for both fitting and evaluation will reduce loss of significance errors on evaluation. Numerical experiments confirm this showing that such an approach can improve overall accuracy by several significant figures.     

Optimal birth control for an age-dependent n-dimensional food chain model

In this paper, we investigate optimal policies for an age-dependent n-dimensional food chain model, which is controlled by fertility. By using Dubovitskii-Milyutin's general theory, the maximum principles are obtained for problems with free terminal states, infinite horizon and target sets, respectively.     

Canonical formalism for parameter-invariant integrals in the Calculus of Variations whose Lagrange functions involve second order derivatives

Summary In a recent paper [4] a general theory of parameter-invariant integrals in the Calculus of Variations whose Lagrangians involve higher derivatives was developed, and in particular a certain canonical formalism for such problems was discussed. From the point of view of applications it was found that this formalism proved inadequate inas-much as the suggested Hamiltonian function did not depend explicitly on the first derivatives of the positional coordinates. In the present note an alternative Hamiltonian function is defined, which gives rise to a new canonical formalism. The latter is less complicated than the formalism suggested in [4] and is more readily applicable to special problems. A brief discussion of the resulting Hamilton-Jacobi theory is given, and in conclusion the method is illustrated explicitly by means of an example of fairly general character.     

Two Families of Approximation Schemes for Delay Systems

Based on the Trotter-Kato approximation theorem for strongly continuous semigroups we develop a general framework for the approximation of delay systems. Using this general framework we construct two families of concrete approximation schemes. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). For the two families we also prove convergence of the adjoint semigroups and uniform exponential stability, properties which are essential for approximation of linear quadratic control problems involving delay systems. The characteristic matrix of the delay system is in both cases approximated by matrices of the same structure but with the exponential function replaced by approximations where Padé fractions in the main diagonal resp. in the diagonal below the main diagonal of the Padé table for the exponential function play an essential role.     

Global analysis of an impulsive delayed Lotka–Volterra competition system

In this paper, a retarded impulsive n-species Lotka–Volterra competition system with feedback controls is studied. Some sufficient conditions are obtained to guarantee the global exponential stability and global asymptotic stability of a unique equilibrium for such a high-dimensional biological system. The problem considered in this paper is in many aspects more general and incorporates as special cases various problems which have been extensively studied in the literature. Moreover, applying the obtained results to some special cases, I derive some new criteria which generalize and greatly improve some well known results. A method is proposed to investigate biological systems subjected to the effect of both impulses and delays. The method is based on Banach fixed point theory and matrix’s spectral theory as well as Lyapunov function. Moreover, some novel analytic techniques are employed to study GAS and GES. It is believed that the method can be extended to other high-dimensional biological systems and complex neural networks. Finally, two examples show the feasibility of the results.     

The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions

We consider a large class of self-adjoint elliptic problems associated with the second derivative acting on a space of vector-valued functions. We present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well known in some special cases) by an abstract rank condition. The second one, though available in general, seems to apply particularly well to a specific boundary condition, the sometimes dubbed anti-Kirchhoff condition in the literature, that arises in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities, in particular some results on the symmetry of the spectrum in either case.     

On the general linear coupled system for diffusion in media with two diffusivities

For the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity, simple uniqueness criteria, and a method of solution of boundary value problems are established. The equations studied retain the so-called cross terms which have been neglected in all previous investigations. Moreover, these equations arise as generalizations of a number of existing theories; for example, heat flow in heterogeneous multicomponent systems, flow of water in fissured rocks and a model of an arms race. The simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants. The solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems. The paper emphasizes the general equations without special reference to particular physical applications or boundary value problems.     

A reduced gradient method for quadratic programs with quadratic constraints and lp-constrained lp-approximation problems

A dual algorithm is developed for solving a general class of nonlinear programs that properly contains all convex quadratic programs with quadratic constraints and l_p-constrained l_p-approximation problems. The general dual program to be solved has essentially linear constraints but the objective function is nondifferentiable when certain variables equal zero. Modifications to the reduced gradient method for linearly constrained problems are presented that overcome the numerical difficulties associated with the nondifferentiable objective function. These modifications permit ‘blocks’ of variables to move to and away from zero on certain iterations even though the objective function is nondifferentiable at points having a block of variables equal to zero.     

The function G λ * as the norm of a Calderón-Zygmund operator

We describe a new approach to one of the quadratic functions in the Littlewood-Paley theory, namely, to the function G _λ *. It is shown that some of its properties can be obtained from the general theory of operators of Calderón-Zygmund type (which, apparently, has not been considered applicable in this context). There are applications to interpolation theory.     

Smoothing Levenberg–Marquardt method for general nonlinear complementarity problems under local error bound

By using the F–B function and smoothing technique to convert the nonlinear complementarity problems to smoothing nonlinear systems, and introducing perturbation parameter μ_k into the smoothing Newton equation, we present a new smoothing Levenberg–Marquardt method for general nonlinear complementarity problems. For general mapping F, not necessarily a P₀ function, the algorithm has global convergence. Each accumulation point of the iterative sequence is at least a stationary point of the problem. Under the local error bound condition, which is much weaker than nonsingularity assumption or the strictly complementarity condition, we get the local superlinear convergence. Under some proper condition, quadratic convergence is also obtained.     

On reductibility of degenerate optimization problems to regular operator equations

We present an application of the p-regularity theory to the analysis of non-regular (irregular, degenerate) nonlinear optimization problems. The p-regularity theory, also known as the p-factor analysis of nonlinear mappings, was developed during last thirty years. The p-factor analysis is based on the construction of the p-factor operator which allows us to analyze optimization problems in the degenerate case. We investigate reducibility of a non-regular optimization problem to a regular system of equations which do not depend on the objective function. As an illustration we consider applications of our results to non-regular complementarity problems of mathematical programming and to linear programming problems.     

Degenerate Evolution Equations in Weighted Continuous Function Spaces, Markov Processes and the Black-Scholes Equation-Part II

In this second part of the paper, through applying semigroup theory procedures, we study initial boundary problems associated with degenerate second-order differential operators of the form Lu(x) ? α(x) u″(x)+β(x)u′(x)+γ(x) u(x) in the framework of weighted continuous function spaces on an arbitrary real interval, when particular boundary conditions are imposed. By using the general results stated in the first part, we show that such operators, frequently occurring in Mathematical Finance, generate positive strongly continuous semigroups, which are, in turn, the transition semigroups associated with suitable Markov processes. Finally, an application to the Black-Scholes equation is discussed, as well.     

Green’s functions for discrete mTH-order problems*

We investigate an mth-order discrete problem with additional conditions, described by m linearly independent linear functionals. We find the solution to this problem and present a formula and the existence condition of Green??s function if the general solution of a homogeneous equation is known. We obtain a relation between Green??s functions of two nonhomogeneous problems. It allows us to find Green??s function for the same equation, but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.     

M/G/∞ with batch arrivals

Let p_∞(n) be the distribution of the number N(∞) in the system at ergodicity for systems with an infinite number of servers, batch arrivals with general batch size distribution and general holding times. This distribution is of importance to a variety of studies in congestion theory, inventory and storage systems. To obtain this distribution, a more general problem is addressed. In this problem, each epoch of a Poisson process gives rise to an independent stochastic function on the lattice of integers, which may be viewed as stochastic impulse response. A continuum analogue to the lattice process is also provided.     

Complexity of Nonlinear Two-Point Boundary-Value Problems

We study upper and lower bounds on the worst-case ε-complexity of nonlinear two-point boundary-value problems. We deal with general systems of equations with general nonlinear boundary conditions, as well as with second-order scalar problems. Two types of information are considered: standard information defined by the values or partial derivatives of the right-hand-side function, and linear information defined by arbitrary linear functionals. The complexity depends significantly on the problem being solved and on the type of information allowed. We define algorithms based on standard or linear information, using perturbed Newton's iteration, which provide upper bounds on the ε-complexity. The upper and lower bounds obtained differ by a factor of log log 1/ε. Neglecting this factor, for general problems the ε-complexity for the right-hand-side functions having r(r⩾2) continuous bounded partial derivatives turns out to be of order (1/ε)^1/r for standard information, and (1/ε)^1/(r+1) for linear information. For second-order scalar problems, linear information is even more powerful. The ε-complexity in this case is shown to be of order (1/ε)^1/(r+2), while for standard information it remains at the same level as in the general case.