Solution of monotone complementarity problems with locally Lipschitzian functions

The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsφ(x)=0 or as the problem of minimizing the merit functionΘ=1/2∥Φ∥ ₂ ² , we extend results which hold for sufficiently smooth functionsF to the nonsmooth case. In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesΘ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.     

Quasipotentials for simple noisy maps with complicated dynamics

The theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems, leading to exponential estimates of invariant probabilities and mean first exit times. In the present article we develop the mathematical foundation of this theory for discrete-time systems, following and extending the work of Freidlin and Wentzell, and Kifer. We discuss strategies for calculating and estimating quasipotentials and show their application to one-dimensionalS-unimodal maps. The method proves to be especially suited for describing the noise scaling behavior of invariant probabilities, e.g., for the map occurring as the limit of the Feigenbaum period-doubling sequence. We show that the method allows statements about the scaling behavior in the case of localized noise, too, which does not originally lie within the scope of the quasipotential formalism.     

Quantum optical ramsey fringes and complementarity

An interferometer in which an atom traverses two identical micromaser cavities in succession is proposed. Depending on the preparation of the cavity fields, the probability for finding the atom in a definite final state displays Ramsey fringes or not. If the initial cavity fields are such that the state of the atom between the cavities can be determined, then the Ramsey fringes disappear, as is required by the principle of complementarity.     

Particle Characterization by Projected Area Determination

The projected areas of non-spherical particles do not represent an unambiguous particle characteristic. Depending on the orientation towards a constant observational direction, different projected areas result. The spectrum of all projected area values of a particle, if determined representatively, gives the probability with which a certain value is obtained by a single measurement. In this work, the frequency distributions of different examples of test objects were both calculated and measured. The objects were a cube, a rectangular parallelepiped and also three model agglomerates consisting of spheres of the same size. Instead of just one projected area, during each measuring procedure three projected areas from three orthogonal directions can be obtained. A mean value is then calculated to reduce the ambiguity of the particle characteristic and enhance the resolution. A suitable measurement set-up is introduced. The results of calculation and measurement are compared for observation from just one direction and also simultaneous observation from three directions. The frequency distributions of the equivalent diameters of the particle projected areas show a characteristic trend of the total curve with remarkable properties. The simultaneous measurement of three values from mutually orthogonal directions and their mean value calculation result in a much narrower distribution. In this case, a non-sphericity factor can additionally be calculated, whose frequency distribution contains information in a characteristic manner about the degree to which the particle shape differs from a sphere.     

Runge–Kutta Methods for Itô Stochastic Differential Equations with Scalar Noise

A general class of stochastic Runge–Kutta methods for Itô stochastic differential equation systems w.r.t. a one-dimensional Wiener process is introduced. The colored rooted tree analysis is applied to derive conditions for the coefficients of the stochastic Runge–Kutta method assuring convergence in the weak sense with a prescribed order. Some coefficients for new stochastic Runge–Kutta schemes of order two are calculated explicitly and a simulation study reveals their good performance.     

Second order Runge–Kutta methods for Stratonovich stochastic differential equations

The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly. AMS subject classification (2000)  65C30, 65L06, 60H35, 60H10