On estimating best approximations of functions defined by differential equations

Finite difference methods are used to estimate the error in approximations to functions defined by differential equations. The problem of minimising the maximum of this error estimate can be solved by linear programming in the linear case, and by a method due to the authors in the nonlinear case. It is shown by examples that this new approach can improve substantially on techniques which minimise the maximum residual in the differential equation, and a convergence result as the mesh spacing tends to zero is given for the linear case.     

Further results on stability of normal regions for linear homogeneous functional equations

In this paper we consider the stability of normal regions on the plane, determined by continuous solutions of the linear homogeneous functional inequality in the case where continuous solutions of the corresponding linear functional equation do not depend continuously on initial conditions.     

Regularization techniques in interior point methods

Regularization techniques, i.e., modifications on the diagonal elements of the scaling matrix, are considered to be important methods in interior point implementations. So far, regularization in interior point methods has been described for linear programming problems, in which case the scaling matrix is diagonal. It was shown that by regularization, free variables can be handled in a numerically stable way by avoiding column splitting that makes the set of optimal solutions unbounded. Regularization also proved to be efficient for increasing the numerical stability of the computations during the solutions of ill-posed linear programming problems. In this paper, we study the factorization of the augmented system arising in interior point methods. In our investigation, we generalize the methods developed and used in linear programming to the case when the scaling matrix is positive semidefinite, but not diagonal. We show that regularization techniques may be applied beyond the linear programming case.     

γ-Active Constraints in Convex Semi-Infinite Programming

In this article, we extend the definition of γ-active constraints for linear semi-infinite programming to a definition applicable to convex semi-infinite programming, by two approaches. The first approach entails the use of the subdifferentials of the convex constraints at a point, while the second approach is based on the linearization of the convex inequality system by means of the convex conjugates of the defining functions. By both these methods, we manage to extend the results on γ-active constraints from the linear case to the convex case.     

PARALLEL STABILIZATION

1.IntroductionLetusconsideranevolutionsystemwhosestateisgivenbythesolutionofaPartialDifferentialEquation(PDE)whichiswritten(formallyforthetimebeing)asIn(1.1),(1.2),whichmaybealinearoranonlinearPDE,ydenotesthestate,andvdenotesthecontrol.TheoperatorA,w...     

Diffusion processes on an interval under linear moment conditions

We discuss a class of linear and nonlinear diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order n. Each choice of n induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of n=1 corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.     

Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion

In this paper, we provide a blow-up mechanism to the modified Camassa–Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics, which can be deduced from tracing the ratio between solution and its gradient. Using the continuity and monotonicity of the solutions, we then extend this blow-up criterion to the case of negative linear dispersion, and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that in the case of non-negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile. In contrast to the Camassa–Holm equation with linear dispersion, the effect of linear dispersion of the modified Camassa–Holm equation on the blow-up phenomena is rather delicate.     

A survey of average case complexity for linear multivariate problems

We survey recent results on the average case complexity for linear multivariate problems. Our emphasis is on problems defined on spaces of functions of d variables with large d. We present the sharp order of the average case complexity for a number of linear multivariate problems as well as necessary and sufficient conditions for the average case complexity not to be exponential in d. Dedicated to the 50th anniversary of the journal. The text was submitted by the authors in English.     

非线性Lipschitz连续算子的定量性质（Ⅳ）──谱理论

本文将有界线性算子谱的定义及若干重要谱性质推广至非线性Ｌｉｐｓｃｈｉｔｚ连续算子，并得到一些有意义的结果．     

Generalized fractional programming: Algorithms and numerical experimentation

Several algorithms to solve the generalized fractional program are summarized and compared numerically in the linear case. These algorithms are iterative procedures requiring the solution of a linear programming problem at each iteration in the linear case. The most efficient algorithm is obtained by marrying the Newton approach within the Dinkelbach approach for fractional programming.     

On nonlinear nuclear operators

A class of operators is introduced and referred to as nonlinear nuclear operators. As in the case of linear nuclear operators this class of nonlinear operators is defined by means of a tensor product of two topological vector spaces. It is then shown that, as in the linear case, a series representation of the operator is valid. The deviation of the nonlinear theory from the linear case is discussed and an example in the framework of generalized stochastic processes is given.     

Completions of implicitly defined linear time varying vector fields

The stabilization of constraints through such techniques as Baumgarte stabilization has been used in the simulation community for some time. This and a number of control problems can be viewed as either extending, or modifying, a vector field off of some manifold. Generally these approaches required the equations to have a special structure. Motivated by numerical simulation there has recently been new progress on doing this stabilization in numerically robust ways for larger classes of systems. In this paper we point out how earlier linear time invariant results do not immediately apply to the linear time varying case and then analyze the linear time varying case.     

Duality gaps in semi-infinite linear programming—an approximation problem

This paper builds upon the relationship between the objective function of a semi-infinite linear program and its constraints to identify a class of semi-infinite linear programs which do not have a duality gap. The key idea is to guarantee the approximation of the primal program by a sequence of linear programs where thenth approximating program is to minimize the objective function subject to the firstn constraints. The paper goes on to show that any program not in the identified class can be linearly perturbed into it with the optimal value of the perturbed program converging to the optimal value of the original program. The results are then extended to the case when an uncountable number of constraints are present by reducing this case to the countable case.     

Time-Irreversibility and Existence and Uniqueness of Solutions of Problems in Linear Viscoelasticity

We study a problem of linear viscoelasticity for the case where the relation between the Cauchy stress and strain tensors is described by a linear integral relation. Theorems on the existence and uniqueness of a solution of the problem are proved.     

Invasion waves in the presence of a mutualist

This paper studies invasion waves in the diffusive competitor–competitor–mutualist model generalizing the two‐species Lotka–Volterra model studied by Weinberger et al. The mutualist may benefit the invading or the resident species producing two different types of invasions. Sufficient conditions for linear determinacy are derived in both cases, and when they hold, explicit formulas for linear spreading speeds of the invasions are obtained by linearizing the model. While in the first case the linear speed is increased by the mutualist, it is unaffected in the second case. Mathematical methods are based on converting the model into a cooperative reaction–diffusion system. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical method for a class of continuous concave programming problems

The problem under consideration consists in maximizing a separable concave objective functional on a class of non-negative Lebesgue integrable functions satisfying a system of linear constraints. The problem is approximated by two sequences of concave separable programming problems with linear constraints. The convergence of the sequences of optimum values of these problems is investigated in the general case and the convergence of the sequences of optimum solutions in a special case. A numerical example is given.     

A compromise solution for the multiobjective stochastic linear programming under partial uncertainty

This paper solves the multiobjective stochastic linear program with partially known probability. We address the case where the probability distribution is defined by crisp inequalities. We propose a chance constrained approach and a compromise programming approach to transform the multiobjective stochastic linear program with linear partial information on probability distribution into its equivalent uniobjective problem. The resulting program is then solved using the modified L-shaped method. We illustrate our results by an example.     

Lyapunov inequality for linear Hamiltonian systems on time scales

The principal aim of this paper is to state and prove some Lyapunov inequalities for linear Hamiltonian system on an arbitrary time scale , so that the well-known case of differential linear Hamiltonian systems and the recently developed case of discrete Hamiltonian systems are unified. Applying these inequalities, a disconjugacy criterion for Hamiltonian systems on time scales is obtained.     

Limit circle invariance for two differential systems on time scales

In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter λ. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value λ₀, then this property is preserved for all complex values λ. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non‐hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors.