Analysis of iterative methods for saddle point problems: a unified approach

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

     

On the Finite-Time Dynamics of Ant Colony Optimization

An analytical framework for investigating the finite-time dynamics of ant colony optimization (ACO) under a fitness-proportional pheromone update rule on arbitrary construction graphs is developed. A limit theorem on the approximation of the stochastic ACO process by a deterministic process is demonstrated, and a system of ordinary differential equations governing the process dynamics is identified. As an example for the application of the presented theory, the behavior of ACO on three different construction graphs for subset selection problems is analyzed and compared for some basic test functions. The theory enables first rough theoretical predictions of the convergence speed of ACO. AMS 2000 Subject classification 68W20, 68W40     

Generalizations of Cayley's -process

In this paper we axiomatize some constructions and results due to Cayley and Hilbert. We define the concept of -process for an arbitrary affine algebraic monoid with zero and unit group . In our situation we show how to produce from the process and for a linear rational representation of a number of elements of the ring of -invariants that is large enough to guarantee its finite generation. Moreover, using complete reducibility, we give an explicit construction of all -processes for reductive monoids.

     

Fast interfacial ionic conduction in nanostructured glass ceramics

The hopping movements of mobile ions in a nanostructured LiAlSiO4 glass ceramic are characterized by time-domain electrostatic force spectroscopy (TDEFS). While the macroscopic conductivity spectra are governed by a single activation energy, the nanoscopic TDEFS measurements reveal three different dynamic processes with distinct activation energies. Apart from the ion transport processes in the glassy and crystalline phases, we identify a third process with a very low activation energy, which is assigned to ionic movements at the interfaces between the crystallites and glassy phase. Such interfacial processes are believed to play a key role for obtaining high ionic conductivities in nanostructured solid electrolytes.     

Universal nature of particle displacements close to glass and jamming transitions

We examine the structure of the distribution of single particle displacements (van Hove function) in a broad class of materials close to glass and jamming transitions. In a wide time window comprising structural relaxation, van Hove functions reflect the coexistence of slow and fast particles (dynamic heterogeneity). The tails of the distributions exhibit exponential, rather than Gaussian, decay. We argue that this behavior is universal in glassy materials and should be considered the analog, in space, of the stretched exponential decay of time correlation functions. We introduce a dynamical model that describes quantitatively numerical and experimental data in supercooled liquids, colloidal hard spheres, and granular materials. The tails of the distributions directly explain the decoupling between translational diffusion and structural relaxation observed in glassy materials.     

Universal reconnection of non-Abelian cosmic strings

We show that local and semilocal strings in Abelian and non-Abelian gauge theories with critical couplings always reconnect classically in collision, by using moduli space approximation. The moduli matrix formalism explicitly identifies a well-defined set of the vortex moduli parameters. Our analysis of generic geodesic motion in terms of those shows right-angle scattering in head-on collision of two vortices, which is known to give the reconnection of the strings.     

Length-scale-dependent relaxation in colloidal gels

We use molecular dynamics computer simulations to investigate the relaxation dynamics of a simple model for a colloidal gel at a low volume fraction. We find that due to the presence of the open spanning network this dynamics shows at low temperature a nontrivial dependence on the wave vector which is very different from the one observed in dense glass-forming liquids. At high wave vectors the relaxation is due to the fast cooperative motion of the branches of the gel network, whereas at low wave vectors the overall rearrangements of the heterogeneous structure produce the relaxation process.     

Photochemistry of refractive index structures in poly(methyl methacrylate) by femtosecond laser irradiation

Femtosecond, subablation threshold photomodification of poly(methyl methacrylate) (PMMA) at 387 nm is explored to enable fabrication of optical components. Volatile fragment analysis (thermal desorption gas chromatography-mass spectrometry) and molecular weight distribution monitoring (size exclusion chromatography) suggest photochemical modification, involving direct cleavage of the polymer backbone and propagation via chain unzipping under formation of monomers, similar to the pyrolytic degradation of PMMA. Waveguides were produced in undoped, clinical-grade PMMA, showing an increased refractive index in the laser focal region (Dnmax=4x10(-3)).