Equivalent Conditions for Local Error Bounds

In this paper we present two classes of equivalent conditions for local error bounds in finite dimensional spaces. We formulate conditions of the first class by using subderivatives, subdifferentials and strong slopes for nearby points outside the referenced set, and show that these conditions actually characterize a uniform version of the local error bound property. We demonstrate this uniformity for the max function of a finite collection of smooth functions, and as a consequence we show that quasinormality constraint qualifications guarantee the existence of local error bounds. We further present the second class of equivalent conditions for local error bounds by using the various limits defined on the boundary of the referenced set. In presenting these conditions, we exploit the variational geometry of the referenced set in a systematic way and unify some existing results in the literature.     

Sensitivity analysis for non-linear programs with linear constraints

We extend the principal sensitivity analysis results for parametric non-lieear programs with linear constraints. In particular, we obtain sufficient conditions for the existence, continuity and differentiability of a locally unique perturbed local minimum. We also show that the local optimal value function is differentiable under these conditions.     

A posteriori error estimation and adaptive mesh-refinement techniques

We analyse three different a posteriori error estimators for elliptic partial differential equations. They are based on the evaluation of local residuals with respect to the strong form of the differential equation, on the solution of local problems with Neumann boundary conditions, and on the solution of local problems with Dirichlet boundary conditions. We prove that all three are equivalent and yield global upper and local lower bounds for the true error. Thus adaptive mesh-refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities. Some numerical examples prove the efficiency of the error estimators and the mesh-refinement techniques.     

Implicit Function Theorem for Systems of Polynomial Equations with Vanishing Jacobian and Its Application to Flexible Polyhedra and Frameworks

 We study the existence problem for a local implicit function determined by a system of nonlinear algebraic equations in the particular case when the determinant of its Jacobian matrix vanishes at the point under consideration. We present a system of sufficient conditions that implies existence of a local implicit function as well as another system of sufficient conditions that guarantees absence of a local implicit function. The results obtained are applied to proving new and classical results on flexibility and rigidity of polyhedra and frameworks.     

On Optimality Conditions for Generalized Semi-Infinite Programming Problems

Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Rückmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian–Fromovitz constraint qualification. Using the ideas of a recent paper of Rückmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian–Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.     

Dynamics in three cells with multiple time delays

We consider systems of delay differential equations representing the models containing three cells with any time-delayed connections. Global stability, delay-independent and delay-dependent local stability are studied, the existence of local and global periodic solutions is investigated. We give the stability conditions, respectively, and show that the local periodic solutions can be extended globally after certain critical values of delay.     

Implicit Function Theorem for Systems of Polynomial Equations with Vanishing Jacobian and Its Application to Flexible Polyhedra and Frameworks

 We study the existence problem for a local implicit function determined by a system of nonlinear algebraic equations in the particular case when the determinant of its Jacobian matrix vanishes at the point under consideration. We present a system of sufficient conditions that implies existence of a local implicit function as well as another system of sufficient conditions that guarantees absence of a local implicit function. The results obtained are applied to proving new and classical results on flexibility and rigidity of polyhedra and frameworks. (Received 13 July 2000; in revised form 11 December 2000)     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

On asymptotic efficiency of exponentiality tests based on Rossberg’s characterization

We construct new tests of exponentiality based on Rossberg’s characterization of the exponential law. We compute limiting distributions of new tests, local Bahadur efficiency for common alternatives, and describe conditions of their local asymptotic optimality. Bibliography: 13 titles.     

An SQP method for optimal control of weakly singular Hammerstein integral equations

We investigate local convergence of an SQP method for nonlinear optimal control of weakly singular Hammerstein integral equations. Sufficient conditions for local quadratic convergence of the method are discussed.     

Positive solutions of certain nonlinear elliptic systems with self-diffusions: nondegenerate vs. degenerate diffusions

We discuss the existence of positive solutions to certain strongly-coupled nonlinear elliptic systems with self-diffusions under homogeneous Dirichlet boundary conditions. Using the global positive coexistence results for predator–prey, competition and symbiotic interactions between two species, sufficient conditions for the positive solutions of the degenerate self-diffusive systems are studied. We also investigate the local behavior, namely, the local existence, uniqueness and stability, of positive solutions for predator–prey and competition interactions. Our method is based on the decoupling technique and bifurcation theory.     

Radiation boundary conditions for time-dependent waves based on complete plane wave expansions

We develop complete plane wave expansions for time-dependent waves in a half-space and use them to construct arbitrary order local radiation boundary conditions for the scalar wave equation and equivalent first order systems. We demonstrate that, unlike other local methods, boundary conditions based on complete plane wave expansions provide nearly uniform accuracy over long time intervals. This is due to their explicit treatment of evanescent modes. Exploiting the close connection between the boundary condition formulations and discretized absorbing layers, corner compatibility conditions are constructed which allow the use of polygonal artificial boundaries. Theoretical arguments and simple numerical experiments are given to establish the accuracy and efficiency of the proposed methods.     

The Local Approximation Theorem in Various Coordinate Systems

We find some sufficient conditions on the local coordinate system of a Carnot–Carathéodory space of low smoothness that ensure Gromov-type estimates on the divergence of local (quasi)metrics. We also obtain these estimates for the canonical coordinate system of the second kind and various mixed coordinate systems.     

Local Properties of Ternary Rings of Operators and Their Linking C *-Algebras

We show that some local properties (such as nuclearity, exactness, and local reflexivity) of ternary rings of operators (TROs) are closely related to the local properties of their linking C^*-algebras. We also show some equivalent conditions for nuclear TROs, and show that Haagerup's decomposition property for completely bounded maps and Pisier's δ-norm can be naturally generalized to TROs.     

A Note on First-Order Sufficient Optimality Conditions for Pareto Problems

We take into consideration the first-order sufficient conditions, established by Jiménez and Novo (Numer. Funct. Anal. Optim. 2002; 23:303–322) for strict local Pareto minima. We give here a more operative condition for a strict local Pareto minimum of order 1.     

Stability for trust-region methods via generalized differentiation

We obtain necessary and sufficient conditions for local Lipschitz-like property and sufficient conditions for local metric regularity in Robinson’s sense of Karush–Kuhn–Tucker point set maps of trust-region subproblems in trust-region methods. The main tools being used in our investigation are dual criteria for fundamental properties of implicit multifunctions which are established on the basis of generalized differentiation of normal cone mappings.     

Local optimality conditions for multicommodity flow problems with separable piecewise convex costs

We consider here a multicommodity flow network optimization problem with non-convex but piecewise convex arc cost functions. We derive complete optimality conditions for local minima based on negative-cost cycles associated with each commodity. These conditions do not extend to the convex non-smooth case.     

A semilinear equation with generalized Wentzell boundary condition

In this paper we consider a semilinear equation with a generalized Wentzell boundary condition. We prove the local well-posedness of the problem and derive the conditions of the global existence of the solution and the conditions for finite time blow-up. We also derive an estimate for the blow-up time.     

On the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient

We study the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient. We obtain sufficient conditions for both global and local (in time) solvability. In the case of local (but not global) solvability, we obtain upper and lower bounds for the existence time of the solution in the form of explicit and quadrature formulas.     

On the absence of local solutions of several evolutionary problems

We consider the existence problem for local (with respect to time) solutions of quasilinear evolutionary partial differential equations and inequalities with singular coefficients and initial conditions. We obtain sufficient conditions for instantaneous blow-up of solutions and show that the results thus obtained cannot be improved in the function class under study.