Higher order numerical differentiation on the Infinity Computer

There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer—the Infinity Computer—able to work numerically with finite, infinite, and infinitesimal numbers. It is proved that the Infinity Computer is able to calculate values of derivatives of a higher order for a wide class of functions represented by computer procedures. It is shown that the ability to compute derivatives of arbitrary order automatically and accurate to working precision is an intrinsic property of the Infinity Computer related to its way of functioning. Numerical examples illustrating the new concepts and numerical tools are given.     

Radiation-induced decomposition of the metal-organic molecule Bis(4-cyano-2,2,6,6-tetramethyl-3,5-heptanedionato)copper(II)

The effects of vacuum ultraviolet radiation on the adsorbed copper center molecule bis(4-cyano-2,2,6,6-tetramethyl-3,5-heptanedionato)copper(II) (or Cu(CNdpm)₂), (C₂₄H₃₆N₂O₄Cu, Cu(II)) was studied by photoemission spectroscopy. Changes in the ultraviolet photoemission spectra (UPS) of Cu(CNdpm)₂, adsorbed on Co(1 1 1), indicate that the ultraviolet radiation leads to decomposition of Cu(CNdpm)₂ and this decomposition is initially dominated by loss of peripheral hydrogen.     

Use of chlorinated carbon and silicon precursors for epitaxial growth of 4H‐SiC at very high growth rates

A possibility to apply the advantages of chlorinated carbon precursors, which had been previously used in low‐temperature epitaxial growth of 4H‐SiC, to achieve very high growth rates at higher growth temperatures was investigated. Silicon tetrachloride was used as the silicon precursor to suppress gas‐phase homogeneous nucleation. The temperature increase from 1300 °C (which is the temperature of the previously reported low‐temperature halo‐carbon epitaxial growth) to 1600 °C enabled an increase of the precursor flow rates and consequently of the growth rate from 5 to more than 100 μm/h without morphology degradation. High quality of the epilayers was confirmed by low‐temperature photoluminescence spectroscopy and time‐resolved luminescence. No evidences of homogeneous nucleation were detected, however, liquid Si droplet formation on the epilayer surface seems to remain a bottleneck at very high growth rate. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gap solitons supported by optical lattices in photorefractive crystals with asymmetric nonlocality

We address the impact of the asymmetric nonlocal diffusion nonlinearity of gap solitons supported by photorefractive crystals with an imprinted optical lattice. We reveal how the asymmetric nonlocal response alters the domains of existence and the stability of solitons originating from different gaps. We find that in such media gap solitons cease to exist above a threshold of the nonlocality degree. We discuss how the interplay between nonlocality and lattice strength modifies the gap soliton mobility.     

Light localization in nonuniformly randomized lattices

We address Anderson localization of light in disordered optical lattices where the disorder strength varies across the transverse direction. Such variation changes the preferred domains where formation of localized eigenmodes is most probable, hence drastically impacting light localization properties. Thus, step-like disorder results in formation of modes with different decay rates at both sides of the interface, while a smoothly varying disorder yields appearance of modes that are extended within weakly disordered domains and rapidly fade away in strongly disordered domains.     

Stable bright and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity

We predict that a photonic crystal fiber whose strands are filled with a defocusing nonlinear medium can support stable bright solitons and also vortex solitons if the strength of the defocusing nonlinearity grows toward the periphery of the fiber. The domains of soliton existence depend on the transverse growth rate of the filling nonlinearity and nonlinearity of the core. Remarkably, solitons exist even when the core material is linear.     

Lipschitz gradients for global optimization in a one-point-based partitioning scheme

A global optimization problem is studied where the objective function f(x)$f\left(x\right)$ is a multidimensional black-box function and its gradient f^′(x)$f^{\prime }\left(x\right)$ satisfies the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant K$K$. Different methods for solving this problem by using an a priori given estimate of K$K$, its adaptive estimates, and adaptive estimates of local Lipschitz constants are known in the literature. Recently, the authors have proposed a one-dimensional algorithm working with multiple estimates of the Lipschitz constant for f^′(x)$f^{\prime }\left(x\right)$ (the existence of such an algorithm was a challenge for 15 years). In this paper, a new multidimensional geometric method evolving the ideas of this one-dimensional scheme and using an efficient one-point-based partitioning strategy is proposed. Numerical experiments executed on 800 multidimensional test functions demonstrate quite a promising performance in comparison with popular DIRECT-based methods.