Non-local approximation of the Mumford-Shah functional

The Mumford-Shah functional, introduced to study image segmentation problems, is approximated in the sense of -convergence by a sequence of non-local integral functionals. Received June 6, 1996 / Accepted July 11, 1996     

On the convergence of Newton's method when estimating higher dimensional parameters

In this paper, we consider the estimation of a parameter of interest where the estimator is one of the possibly several solutions of a set of nonlinear empirical equations. Since Newton's method is often used in such a setting to obtain a solution, it is important to know whether the so obtained iteration converges to the locally unique consistent root to the aforementioned parameter of interest. Under some conditions, we show that this is eventually the case when starting the iteration from within a ball about the true parameter whose size does not depend on n. Any preliminary almost surely consistent estimate will eventually lie in such a ball and therefore provides a suitable starting point for large enough n. As examples, we will apply our results in the context of M-estimates, kernel density estimates, as well as minimum distance estimates.     

Exciton-exciton interaction in coupled quantum wells

The exciton-exciton interaction is investigated for spatially indirect excitons in coupled quantum wells. The Hartree-Fock and Heitler-London approaches are improved by a full two-exciton calculation including the van der Waals effect. Using these potentials for the singlet and triplet channel, the two-body scattering matrix is calculated and employed to derive a modified relation between exciton density and blue shift. Such a relation is of central importance for gauging exciton densities on the way toward Bose condensation.     

Dynamic supercoiling bifurcations of growing elastic filaments

Certain bacteria form filamentous colonies when the cells fail to separate after dividing. In Bacillus subtilis, Bacillus thermus, and Cyanobacteria, the filaments can wrap into complex supercoiled structures as the cells grow. The structures may be solenoids or plectonemes, with or without branches in the latter case. Any microscopic theory of these morphological instabilities must address the nature of pattern selection in the presence of growth, for growth renders the problem nonautonomous and the bifurcations dynamic. To gain insight into these phenomena, we formulate a general theory for growing elastic filaments with bending and twisting resistance in a viscous medium, and study an illustrative model problem: a growing filament with preferred twist, closed into a loop. Growth depletes the twist, inducing a twist strain. The closure of the loop prevents the filament from unwinding back to the preferred twist; instead, twist relaxation is accomplished by the formation of supercoils. Growth also produces viscous stresses on the filament which even in the absence of twist produce buckling instabilities. Our linear stability analysis and numerical studies reveal two dynamic regimes. For small intrinsic twist the instability is akin to Euler buckling, leading to solenoidal structures, while for large twist it is like the classic writhing of a twisted filament, producing plectonemic windings. This model may apply to situations in which supercoils form only, or more readily, when axial rotation of filaments is blocked. Applications to specific biological systems are proposed.     

III-V compound semiconductor (001) surfaces

There has been renewed interest in the structure of III-V compound semiconductor (001) surfaces caused by recent experimental and theoretical findings, which indicate that geometries different from the seemingly well-established dimer models describe the surface ground state for specific preparation conditions. I review briefly the structure information available on the (001) surfaces of GaP, InP, GaAs and InAs. These data are complemented with first-principles total-energy calculations. The calculated surface phase diagrams are used to explain the experimental data and reveal that the stability of specific surface structures depends largely on the relative size of the surface constituents. Several structural models for the Ga-rich GaAs (001)(4×6) surface are discussed, but dismissed on energetic grounds. I discuss in some detail the electronic properties of the recently proposed cation-rich GaAs (001)ζ(4×2) geometry. Received: 18 May 2001 / Revised version: 23 July 2001 / Published online: 3 April 2002     

Thermal and chemical nanofractal relaxation

We studied shape relaxation of nano-fractal islands, during annealing, after their growth from antimony cluster deposition on graphite surface. Annealing at 180^°C shows evidence of an increase of the fractal branch width with time followed by branch fragmentation, without changing the fractal dimension. The time evolution of the width of the arm suggests the surface self-diffusion mechanism as the main relaxation process. With Monte Carlo simulations, we confirmed the observed behavior. Comparison is done with our previous results on fragmentation of nano-fractal silver islands when impurity added to the incident cluster promotes rapid fragmentation by surface self-diffusion enhancement [1].     

A model for the interaction of near-infrared laser pulses with metal powders in selective laser sintering

A thermal model of the interaction of pulsed near-infrared laser radiation from a Nd:YAG laser was made, taking the measured powder properties such as reflectance, optical penetration depth and thermal conductivity into account. It allows an estimation of the evolution of two different temperatures: the average temperature of the powder (taken over the grains in a volume given by the laser beam diameter and the optical penetration depth) and the temperature distinction within a single grain. It showed that in pulsed mode consolidation can be achieved at much lower average power as the surface of the powder particles are molten but their cores remain at nearly room temperature. This leads to a much lower average temperature and therefore a dramatic decrease in residual thermal stresses in the finished piece. The results of the model were experimentally tested and confirmed. Received: 26 July 2001 / Accepted: 23 November 2001 / Published online: 23 January 2002     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

A comparative study on the interaction between tin(II) porphyrin and selected group 8B metals for potential application as reduction catalysts

We study the interaction between tin(II) porphyrin (SnPor) with platinum and non-precious Group 8B metals (iron, cobalt and nickel) by density functional theory and discuss the electronic properties of the resulting products. We also model the interaction of the resulting compounds with water where applicable. Our studies indicate that, SnPor-Ni possesses electronic properties similar to SnPor-Pt, suggesting that it may possess similar photocatalytic properties for reduction reactions, such as converting water to hydrogen gas.