The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

     

Enhanced x-ray phase determination by three-beam diffraction

For decades, solving the phase problem of x-ray scattering has been a goal that, in principle, could be achieved by means of n-beam diffraction (n-BD). However, the phases extracted by the actual n-BD phasing techniques are not very precise, mainly due to systematic errors that are difficult to estimate. We present an innovative theoretical approach and experimental procedure that, combined, eliminate two major sources of error. It is a high precision phasing technique that provides the triplet-phase angle with an error of about 2 degrees.     

Precise Coupling Terms in Adiabatic Quantum Evolution

It is known that for multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. For a family of two-state systems with real-symmetric Hamiltonian we construct such a superadiabatic representation and explicitly determine the asymptotic behavior of the exponentially small coupling term. First order perturbation theory in the superadiabatic representation then allows us to describe the time-development of exponentially small adiabatic transitions. The latter result rigorously confirms the predictions of Sir Michael Berry for our family of Hamiltonians and slightly generalizes a recent mathematical result of George Hagedorn and Alain Joye.submitted 29/06/04, accepted 14/08/04     

Physics of evolution: Selection without fitness

Traditionally evolution is seen as a process where from a pool of possible variations of a population (e.g. biological species or industrial goods) a few variations get selected which survive and proliferate, whereas the others vanish. Survival probabilities and proliferation rates are typically associated with the ‘fitness’ of particular variations. In this paper we argue that the notion of fitness is an a posteriori concept, in the sense that one can assign higher fitness to species that survive but one can generally not derive or predict fitness per se. Proliferation rates can be measured, whereas fitness landscapes, i.e. the inter-dependence of proliferation rates, cannot. For this reason we think that in a physical theory of evolution such notions should be avoided. In this spirit, here we propose a random matrix model of evolution where selection mechanisms are encoded in interaction matrices of species, thereby extending the previous work of ours by a control parameter describing suppressors in the system. We are able to recover some key facts of evolution dynamics endogenously, such as punctuated equilibrium, i.e. the existence of intrinsic large extinction events, and, at the same time, periods of dramatic diversification, as known e.g. from the fossil record. Further, we comment on two fundamental technical problems of a ‘physics of evolution’, the non-closedness of its phase space and the problem of co-evolving boundary conditions, apparent in all systems subject to evolution.     

From holes to sponge at irradiated Ge surfaces with increasing ion energy—an effect of defect kinetics?

We show that hole patterns and sponge-like layers at irradiated Ge surfaces originate from the same driving force, namely the kinetics of ion beam induced defects in the amorphous Ge surface layer. Ge hole patterns reported earlier for irradiation with low energy (5 keV) Ga⁺ ions were reproduced for low energy Bi⁺ but also for Ge⁺ self-irradiation, which proves that the dominating driving force for morphology evolution cannot originate from the implanted impurities. At higher ion energies the well-known formation of sponge-like Ge surface layers after heavy ion irradiation was found for Bi⁺ irradiation and Ge⁺ self-irradiation, also. The transition from smooth surfaces via hole patterns to sponge-like morphologies with increasing ion energies was studied in detail. A model based on the kinetics of ion beam induced defects was developed and implemented in 3D kinetic Monte Carlo simulations, which reproduce the transition from hole patterns to sponge-like layers with increasing ion energy.     

Parallel information-based complexity

We develop the theory of information-based complexity from a parallel point of view. For a model of computation with p processors, each being capable of arithmetic operations and decisions, we analyze the complexity of function approximation, numerical integration, and solution of Fredholm integral equations. We obtain tight bounds on the complexity, considered as a function of three variables simultaneously: the number of processors, the required precision, and (in the case of approximation and integral equations) the number of points, in which the approximate solution is to be determined.     

The Role of Information in Managing Interactions from a Multifractal Perspective

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”.     

Reduction of continuous symmetries of chaotic flows by the method of slices

We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a ‘slice’ defined by minimizing the distance to a single generic ‘template’ intersects the group orbit of every point in the full state space. Global symmetry reduction by a single slice is, however, not natural for a chaotic/ turbulent flow; it is better to cover the reduced state space by a set of slices, one for each dynamically prominent unstable pattern. Judiciously chosen, such tessellation eliminates the singular traversals of the inflection hyperplane that comes along with each slice, an artifact of using the templates local group linearization globally. We compute the jump in the reduced state space induced by crossing the inflection hyperplane. As an illustration of the method, we reduce the SO (2) symmetry of the complex Lorenz equations.