Flatness in analytic mappings I. On an announcement of Hironaka

We present a stratification by “normal flatness” associated to an analytic mapping, analogous to Hironaka's classical result for analytic spaces. Our construction is based on a generic normal flatness theorem for mappings, proved using techniques concerning the variation of modules of meromorphically parametrized formal power series [1]. The existence of such a stratification was announced by Hironaka [13], but the other claims made in [13] are false. Counterexamples are also presented here.     

Existence and Stability for Spherical Crystals Growing in a Supersaturated Solution

We consider the growth of a spherical crystal in a supersaturatedsolution. In the first part, existence and uniqueness resultsfor radially symmetric growth are obtained, provided that thesupersaturation is not too large; conversely, when the far-fieldsupersaturation exceeds a critical value, it is shown that theradially symmetric solution ceases to exist in finite time.In the second part, we examine the linear stability of a radiallysymmetric similarity solution (in which the radius grows ast?) to shape perturbations. The results are compared with previousquasi-static analyses, and, in particular, the critical radiusat which the crystal becomes unstable is found to be largerfor small supersaturations, but smaller for large supersaturations,than those predicted by the quasi-static analysis     

On the distribution of the number of real zeros of a random polynomial

In this note, we find the distibution of the number of real zeros of a random polynomial. We also derive a formula for the expected number of complex zeros lying in a given domain of the complex plane. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 69–79.     

Duality for Finite Hopf Algebras Explained by Corings

We give a coring version for the duality theorem for actions and coactions of a finitely generated projective Hopf algebra. We also provide a coring analogue for a theorem of H.-J. Schneider, which generalizes and unifies the duality theorem for finite Hopf algebras and its refinements. This paper was written while the first author visited the Mathematics Departments of Syracuse University and California State University Dominguez Hills. He would like to thank both departments for their hospitality.     

Finite-difference time-domain simulation of scattering from objectsin continuous random media

A three-dimensional (3D) finite-difference time-domain (FDTD) scheme is introduced to model the scattering from objects in continuous random media. FDTD techniques have been previously applied to scattering from random rough surfaces and randomly placed objects in a homogeneous background, but little has been done to simulate continuous random media with embedded objects where volumetric scattering effects are important. In this work, Monte Carlo analysis is used in conjunction with FDTD to study the scattering from perfectly electrically conducting (PEC) objects embedded in continuous random media. The random medium models under consideration are chosen to be inhomogeneous soils with a spatially fluctuating random permittivities and prescribed correlation functions. The ability of frequency averaging techniques to discriminate objects in this scenarion is also briefly investigated. The simulation scheme described in this work can be adapted and used to help in interpreting the scattered field data from targets in random environments such as geophysical media, biological media, or atmospheric turbulence     

On the problem of calculating the optical film parameters from the measured transmission spectra

The procedure for calculating the dispersion of the refractive and absorption indices and thickness of an optical film, based on measurements of spectral dependence of the transmittance, is described. The film parameters are calculated using a solution of the approximation problem. The effect of the film model on the calculated values of the refractive and absorption indices is analyzed.     

Geometrical method for determining the surface damage threshold for femtosecond laser pulses

A new geometrical method to determine the surface damage threshold for ultrashort pulses is presented. It consists in the formation of a surface damage profile by a movement of the sample across the laser beam focus. A single measurement of the maximum transversal dimension of this damage profile, which depends solely on the laser beam power, is used to calculate the local damage threshold intensity. It is also theoretically shown that the damage in the transverse dimension can be controlled under the diffraction-limited spot.     

Global stability of the Armstrong-Frederick model with periodic biaxial inputs

The paper is concerned with the study of plasticity models described by differential equations with stop and play operators. We suggest sufficient conditions for the global stability of a unique periodic solution for the scalar models and for the vector models with biaxial inputs of a particular form, namely the sum of a uniaxial function and a constant term. For another class of simple biaxial inputs, we present an example of the existence of unstable periodic solutions. The paper was written during the research stay of D. Rachinskii at the Technical University Munich supported by the research fellowship from the Alexander von Humboldt Foundation. His work was partially supported by the Russian Science Support Foundation, Russian Foundation for Basic Research (Grant No. 01-01-00146, 03-01-00258), and the Grants of the President of Russia (Grant No. MD-87.2003.01, NS-1532.2003.1). The support is gratefully acknowledged.     

Performance engineering and topological design of metro WDM opticalnetworks using computer simulation

This paper demonstrates the use of computer simulation for topological design and performance engineering of transparent wavelength-division multiplexing metropolitan-area networks. Engineering of these networks involves the study of various transport-layer impairments such as amplifier noise, component ripple, chirp/dispersion, optical crosstalk, waveform distortion due to filter concatenation, fiber nonlinearities, and polarization effects. A computer simulation methodology composed of three main simulation steps is derived and implemented. This methodology obtains performance estimations by applying efficient wavelength-domain simulations on the entire network topology, followed by time-/frequency-domain simulations on selected paths of the network and finally Q-budgeting on an identified worst case path. The above technique provides an efficient tool for topological design and network performance engineering. Accurate simulation models are presented for each of the performance impairments, and the computer simulation methodology is used for the design and engineering of a number of actual metro network architectures