A counterexample to a global optimization algorithm

A counterexample to an algorithm of Dien (1988) for solving a minimization problem with a quasiconcave objective function and both linear and a reverse-convex constraint shows that this algorithm needn't lead to a solution of the given problem.     

A zero-entropy mixing transformation whose product with itself is loosely Bernoulli

A zero-entropy mixing transformationT is constructed such thatT×T is loosely Bernoulli (LB). Previously known examples were not mixing. The construction is then generalized to yield a zero-entropy mixing transformationT such that then-fold productT × … ×T is LB for each positive integern. Furthermore, a flow with the same properties is obtained.     

On the convergence of a regularizing Levenberg–Marquardt scheme for nonlinear ill-posed problems

In this note, we study the convergence of the Levenberg–Marquardt regularization scheme for nonlinear ill-posed problems. We consider the case that the initial error satisfies a source condition. Our main result shows that if the regularization parameter does not grow too fast (not faster than a geometric sequence), then the scheme converges with optimal convergence rates. Our analysis is based on our recent work on the convergence of the exponential Euler regularization scheme (Hochbruck et al. in Inverse Probl 25(7):075009, 2009).