A field theoretic approach that is based on the method of characteristics is outlined to prove the optimality of a synthesis of controlled trajectories that contains chattering arcs. Since the corresponding controls are not piecewise continuous they switch infinitely many times on finite intervals the classical argument by Boltyansky to prove differentiability of the associated value function i...

A class of semismooth Newton methods for unilaterally constrained variational problems modelling cracks under a non-penetration condition are introduced and investigated. On the continuous level, a penalization technique is applied which allows to argue generalized differentiability of the nonlinear mapping associated to its first order optimality characterization. It is shown that the correspo...

Recently, Huang and Ng presented second-order sufficient conditions for error bounds of continuous and Gâteaux differentiable functions in Banach spaces. Wu and Ye dropped the assumption of Huang and Ng on Gâteaux differentiability but required the space to be a Hilbert space. We carry on this research in two directions. First we extend Wu and Ye’s result to some non-Hilbert spaces; second, sam...

A quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere, but differentiable nowhere. This lack of differentiability has been used as an argument to deny the general validity of Bohmian mechanics (and other trajectory–based approaches) in providing a complete interpretation of quantum mechanics. Here, this assertion is overcome by means of a formal extension o...

Here, we prove Theorem 1 from Section 3, i.e., the equivalence between the length of any given curve under the geodesic distance δg and the Stein metric δS up to scale of 2 √ 2. The proof of this theorem follows several steps. We start with the definition of curve length and intrinsic metric. Without any assumption on differentiability, let (M, d) be a metric space. A curve inM is a continuous ...

We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equa...

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measuretheoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the exist...

2 Finite-Dimensional Optimization 5 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We study numerically and analytically the properties of the stationary state of a particle moving under the influence of an electric field E in a two dimensional periodic Lorentz gas with the energy kept constant by a Gaussian thermostat. Numerically the current appears to be a continuous function of E whose derivative varies very irregularly, possibly in a discontinuous manner. We argue for th...

We perform a bifurcation analysis of the mathematical model of Jones and Kompala [K.D. Jones, D.S. Kompala, Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures, J. Biotechnol. 71 (1999) 105-131]. Stable oscillations arise via Andronov-Hopf bifurcations and exist for intermediate values of the dilution rate as has been noted from experiments previ...