Penalty Algorithms in Hilbert Spaces

We analyze the classical penalty algorithm for nonlinear programming in Hilbert spaces and obtain global convergence results, as well as asymptotic superlinear convergence order. These convergence results generalize similar results obtained for finite-dimensional problems. Moreover, the nature of the algorithms allows us to solve the unconstrained subproblems in finite-dimensional spaces.     

Remarks on ideals in join-semilattices

A set of results analogous to the corresponding results in lattices is derived for neutral ideals in join-semilattices, for congruence relations generated by neutral ideals in join-semilattices, and for permutable congruence relations generated by ideals in modular join-semilattices.     

Stability results for Jungck-type iterative processes in convex metric spaces

In this paper, we obtain some stability results in complete convex metric spaces for nonselfmappings satisfying certain general contractivity condition. We prove our results for Jungck-Mann and Jungck-Ishikawa iterations. Our results extend several stability results in the literature.     

Pseudocomplementation and minimal prime ideals in semirings

In Section 2 of the present paper, we introduce the concept of pseudocomplementation for semirings and show the semiring version of some known results in lattice theory. We also introduce semirings with pc-functions and prove some interesting results for minimal prime ideals of such semirings. In Section 3, some classical results for minimal prime ideals in ring theory are generalized in the context of semiring theory.     

Pseudo-invexity and duality in nonlinear programming

The purpose of this paper is to study various duality results in nonlinear programming for pseudo-invex functions. Such results were known in the literature for invex functions.     

On stability in finite models

We search for a set-up in which results from the theory of infinite models hold for finite models. As an example we prove results from stability theory. Received: 23 February 1998     

Generalized projections in Banach algebras

In this note we investigate generalized projections in Banach algebras. Our results generalize results obtained for bounded linear operators on Hilbert spaces.     

Ruin probabilities allowing for delay in claims settlement

In this paper we use martingale techniques to derive upper bounds for the probability of ruin for a risk process. The important difference between our results and previous results in this area is that our model for the risk process explicitly allows for delay in claims settlement.     

Approximate duality of g-frames in Hilbert spaces

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

Semidefinite optimization in discrepancy theory

Recently, there have been several new developments in discrepancy theory based on connections to semidefinite programming. This connection has been useful in several ways. It gives efficient polynomial time algorithms for several problems for which only non-constructive results were previously known. It also leads to several new structural results in discrepancy itself, such as tightness of the so-called determinant lower bound, improved bounds on the discrepancy of the union of set systems and so on. We will give a brief survey of these results, focussing on the main ideas and the techniques involved.     

Equilibrium Problems in the Quasimonotone Case

Existence results for quasimonotone vector equilibrium problems and quasimonotone vector variational inequalities are obtained starting from an existence result for a scalar equilibrium problem involving two quasimonotone bifunctions. These results are established under weaker conditions than in previous works.     

Pulsating entry-flow in a plane channel

To obtain results for the title problem, the time-dependent Navier-Stokes equations have been solved numerically. Axial-velocity profiles at various distances from the entrance of the channel are shown for a number of points in time during one period of oscillation. Further some results for the time-dependent inlet length are presented.     

Majorizing sequences for iterative procedures in Banach spaces

Newton-like methods are often used for solving nonlinear equations. In the present paper, we introduce very general majorizing sequences for Newton-like methods. Then, we provide semi-local convergence results for these methods. The new convergence results can be weaker than in earlier studies. These new results are illustrated by several numerical examples and special cases of Newton-like methods, for which the older convergence conditions do not hold but for which our weaker convergence conditions are satisfied.     

Submanifolds in Cauchy Riemann Geometry

In this paper I would like to make a report on the results about hypersurfaces in the Heisenberg group and invariant curves and surfaces in CR geometry. The results are contained in the papers [8, 9, 16] and [14]. Besides, I would also report on the results about the strong maximum principle for a class of mean curvature type operators in [10].     

On nonlinear differential equations in Hilbert spaces

In a recent paper of the same title, L. A. Medeiros has considered the question of uniqueness for the Cauchy problem for ordinary differential equations in a complex Hilbert space. Section 1 contains a discussion of Medeiros' results, and provides a motivation for the improved uniqueness results in Section 2     

Dynamical Systems and Adaptive Timestepping in ODE Solvers

Initial value problems for ODEs are often solved numerically using adaptive timestepping algorithms. These algorithms are controlled by a user-defined tolerance which bounds from above the estimated error committed at each step. We formulate a large class of such algorithms as discrete dynamical systems which are discontinuous and of higher dimension than the underlying ODE. By assuming sufficiently strong finite-time convergence results on some neighbourhood of an attractor of the ODE we prove existence and upper semicontinuity results for a nearby numerical attractor as the tolerance tends to zero.This assumption of sufficiently strong finite-time convergence results is then examined for adaptive algorithms that use a pair of explicit Runge-Kutta methods of different order to estimate the one-step error. For arbitrary Runge-Kutta pairs the necessary finite-time convergence results fail to hold on a set of points in the phase space that includes all the equilibria of the ODE. Therefore, in general, the asymptotic convergence results cannot be applied to attractors containing equilibria. However, for a particular class of Runge-Kutta pairs, the finite-time convergence results can be strengthened to include neighbourhoods of equilibrium points for which the Jacobian is invertible.     

Measures and Integrals in Conditional Set Theory

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces.     

Further Algebraic Results in the Theory of Balance

In an article published in 1970 in the French journal Mathématiques et Sciences Humaines , Flament described a number of mathematical results pertaining to an algebraic formulation of the theory of balance in signed graphs. This paper describes some additional results based on those discussed by Flament. These newer results are especially useful for the development of algorithms for calculating the line index of balance in the study of specific networks arising in psychological and sociological research.     

Subgradients of marginal functions in parametric mathematical programming

In this paper we derive new results for computing and estimating the so-called Fréchet and limiting (basic and singular) subgradients of marginal functions in real Banach spaces and specify these results for important classes of problems in parametric optimization with smooth and nonsmooth data. Then we employ them to establish new calculus rules of generalized differentiation as well as efficient conditions for Lipschitzian stability and optimality in nonlinear and nondifferentiable programming and for mathematical programs with equilibrium constraints. We compare the results derived via our dual-space approach with some known estimates and optimality conditions obtained mostly via primal-space developments.