Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation: An Improvement

In this paper, we consider a discrete delay problem with negative feedback x(t)=f(x(t), x(t−1)) along with a certain family of time discretizations with stepsize 1/n. In the original problem, the attractor admits a nice Morse decomposition. We proved in (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36-78) that the discretized problems have global attractors. It was proved in (T. Gedeon and K. Mischaikow, 1995, J. Dynam. Differential Equations7, 141-190) that such attractors also admit Morse decompositions. In (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36-78) we proved certain continuity results about the individual Morse sets, including that if f(x, y)=f(y), then the individual Morse sets are upper semicontinuous at n=∞. In this paper we extend this result to the general case; that is, we prove for general f(x, y) with negative feedback that the Morse sets are upper semicontinuous.     

3D Study of the Morphology and Dynamics of Zeolite Nucleation

The principle aspects and constraints of the dynamics and kinetics of zeolite nucleation in hydrogel systems are analyzed on the basis of a model Na‐rich aluminosilicate system. A detailed time‐series EMT‐type zeolite crystallization study in the model hydrogel system was performed to elucidate the topological and temporal aspects of zeolite nucleation. A comprehensive set of analytical tools and methods was employed to analyze the gel evolution and complement the primary methods of transmission electron microscopy (TEM) and nuclear magnetic resonance (NMR) spectroscopy. TEM tomography reveals that the initial gel particles exhibit a core–shell structure. Zeolite nucleation is topologically limited to this shell structure and the kinetics of nucleation is controlled by the shell integrity. The induction period extends to the moment when the shell is consumed and the bulk solution can react with the core of the gel particles. These new findings, in particular the importance of the gel particle shell in zeolite nucleation, can be used to control the growth process and properties of zeolites formed in hydrogels.     

Focussed surface acoustic waves on an anisotropic crystal, studied by optical probing

Efficient focussing of surface acoustic waves has been achieved using a properly shaped gold film deposit on the −22.3° rotated Y-cut surface of quartz. The acoustic wave field was studied with laser probing techniques. A more than threefold increase in intensity and a tenfold decrease in beam width was observed at the focal point. The focussing action was obtained with the elastic wave equivalent of the Fresnel phase-reversal zone plate of optics. The multiple foci of this device allowed a simultaneous generation of acoustic waves in nearly all directions on the surface. Consequently, the surface wave velocity anisotropy could be determined completely. The experimental results are in very good agreement with the calculated velocity anisotropy. Electromagnetic diffraction theory is adopted to the two dimensional anisotropic system to analyse the performance of the focussing device.     

Volume fluctuations in potassium silicate glass studied by molecular dynamics

Potassium-silicate glass, 0.05K₂O·0.95SiO₂, was prepared by molecular dynamics under three different regimes (cooling rates, simulating box sizes) to study their influence to temperature regimes of volume fluctuations. Partial volume fluctuations (VF) were introduced with the help of Voronoi polyhedra tessellation. Dynamic and static fluctuations were suggested as a measure for time- and space-related fluctuations. Separation of the dynamic and static VFs anticipates glass transition. The glass transition was related with the change of the temperature course of the static VFs. The dynamic VFs changed their temperature regimes well below the glass transition and the transition temperature was related with the change of the transport regime. Glass transition cages the atoms but Voronoi polyhedra are definitely shaped well below the glass transition temperature.     

Low energy and low dose electron irradiation of potassium–lime–silicate glass investigated by XPS. I. Surface composition

X-ray induced photoelectron spectroscopy has been used to study the influence of low-energy electron beam on the pristine potassium–lime–silicate glass surface prepared by fracturing in situ under ultrahigh vacuum. Relatively low-energy electron beam of 1600 eV with low-electron beam current density of 0.02–0.22 A/m² and low-electron dose of 29–5200 C/m² was used. The expected modified near-surface region thickness is in this case comparable with the mean sampling depth of the analytical tool. Therefore, possible changes and modifications due to electron irradiation could be recorded with high sensitivity. The freshly fractured glass surface was found to be significantly enriched with potassium, and slightly with calcium. As a consequence of the lowest electron dose irradiation used, the potassium signal substantially increased by a factor 1.24 relative to the value found for the fresh surface. For higher doses used, the potassium signal continuously increased with the dose to a maximum and decreased thereafter. This variation was accompanied with the qualitative opposite behaviour of calcium signal. The concentrations of the other elements present in the glass, oxygen and silicon, varied only slightly with the electron dose. They can be considered to be constant within experimental uncertainty. In agreement with experimental results, a model assuming mobility of only two most mobile cations, potassium and calcium, was suggested. The models assuming one layer and two layers on the bulk were developed. Their results reproduce well experimental findings: (i) the formation of a potassium-rich surface layer, and (ii) the opposite-like signal variation of calcium in comparison with potassium.     

Chemical state of Na and K ions in Na10K5 silicate glass under electron irradiation investigated by XPS with the aid of the line shape analysis

The Na? K (Na10K5) silicate glass, unirradiated and electron irradiated (electron dose from 25 to 8239 Cm^?2) is investigated using XPS. The measurements are performed in the angular‐resolved ADES‐400 spectrometer using AlKα X‐ray radiation and an electron beam of energy 2 keV. Owing to surface charging and ambiguity of identification of the atomic oxidized chemical states, the line shapes of selected XPS transitions are analyzed with the aid of the pattern recognition (PR) method. This method is based on a distance measure and deals with spectra representation as vectors in the n‐dimensional space. The algorithm presented, called the fuzzy k‐nearest neighbor (fkNN) rule, allows for identification of ambiguous vectors with the membership vectors described by classes membership probabilities. Under electron irradiation, the Na and K content in a surface region undergoesincrease and then slow systematic decrease. The line shape analysis indicates difficult classification of XPS spectra recorded for unirradiated and irradiated glass, especially for Na 1s transition. The chemical state of Na is a mixture of elemental and oxidized form and remains unchanged for all electron doses. Larger changes in the chemical form are observed for the K atom. In an unirradiated silicate glass, a mixture of elemental and oxide form is observed with increasing content of oxide under irradiation. The alkali atoms, Na and K, exhibit a migration effect. Comparison of PR and fitting results indicates better reliability and accuracy of the PR method. Copyright © 2008 John Wiley & Sons, Ltd.     

The method of homogeneous solutions in an axisymmetric problem of statics for a finite cylinder with homogeneous conditions on the ends

Using the property of Papkovich generalized orthogonality of eigenfunctions, we develop a method of satisfying the boundary conditions on the lateral surface of a cylinder. The stresses and displacements in a finite cylinder with homogeneous conditions on the ends are represented in terms of the axial displacement. The solution is constructed as an expansion in a series of eigenfunctions of the corresponding homogeneous boundary-value problem. We find a class of boundary conditions that admits a solution of the problem without reduction to an infinite system of algebraic equations. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 135–139.     

Oscillations in multi-stable monotone systems with slowly varying feedback

The study of dynamics of gene regulatory networks is of increasing interest in systems biology. A useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems. Key features of the dynamics of the original system are then deduced from the input-output characteristics of the open loop system and the sign of the feedback. This paper extends these results, showing how to use the same framework of input-output systems in order to prove existence of oscillations, if the slowly varying strength of the feedback depends on the state of the system.     

Multi-valued characteristics and Morse decompositions

A rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area.We combine a theory of monotone input-output systems with a classical theory of Morse decompositions in the context of ordinary differential equations models of biochemical reactions. We show that a multi-valued input-output characteristic can be used to define non-trivial Morse decompositions which provide information about a global structure of the attractor. The previous work on input-output characteristics is shown to apply locally to individual Morse sets and is seamlessly incorporated into our global theory.We apply our tools to a model of cell cycle maintenance. We show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways: it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions.     

Monte Carlo correction programme for PC

A Monte Carlo correction program for quantitative microanalysis on PC computer is introduced in this paper. The elastic scattering is described by the screened Rutherford cross section. Instead of computing the energy loss according to the actual path between two scatterings we have defined the Bethe inelastic cross section determined by the Bethe-slowing-down approximation. It is assumed that it causes no angular departure of the scattered electron. In the second model we took into account the angular dependence of inelastic scattering assuming that the primary electron interacts with quasi-free atom electrons. On the basis of these two models analytical Monte Carlo programmes were developed and experimentally tested on some oxide glass. Our results are fully comparable to those obtained by ten world microprobe laboratories using classical ZAF correction or Bence-Albee methods. We have found that introducing angular part of the inelastic cross section analytical results did not significantly change. All of our results were carried out for bulk specimens but extending it to layers is under the development.