The Conley index for fast-slow systems II: Multidimensional slow variable

We use the Conley index theory to develop a general method to prove existence of periodic and heteroclinic orbits in a singularly perturbed system of ODEs. This is a continuation of the authors' earlier work [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470] which is now extended to systems with multidimensional slow variables. The key new idea is the observation that the Conley index in fast-slow systems has a cohomological product structure. The factors in this product are the slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast-slow systems with one-dimensional slow flow [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470].     

Microanalysis of Glass Containing Alkali Ions

 The common problems connected with alkali ion migration during EPMA were studied on glasses containing nearly all possible alkali ions (Na, K, Rb, Cs). Binary silica glasses were prepared by melting from a very pure batch in Pt crucible. The glasses were carefully polished using alcohol to prevent surface corrosion by water and they were stored in vacuum. The specimens were coated with carbon layers approximately 30-nm thick and exposed to a 50- keV electron beam of 100 μm diameter. It was found that all alkali ions migrate under the electron beam, but the rate of the migration depends on the current density. The decay curves (characteristic X-ray intensity versus time) are similar in shape in all cases. The decay curve shows two transport regimes, the first being linear-like, the second being the exponential-like. The first transport regime busts into the rapid alkali migration after a time known as the incubation period. The period is in general longer for the larger-alkali ions size. It was found that even large rubidium and caesium ions migrate inside the glass with the same mechanism as sodium and potassium ions. While for K, Rb, and Cs ions the incubation periods were observed under the suitable experimental conditions, binary glass containing Na exhibits no observable incubation period. Except for the binary Na₂O + SiO₂ glass, the suitable experimental conditions for reliable quantitative EPMA can be found.     

Connectivity of Q-species in binary sodium-silicate glasses

Series of sodium-silicate glasses were simulated by molecular dynamics. Glass structures were described in frame of Q-species. Connectivity among different Q-species decouples structure into alkali-associated and alkali-migration-blocking parts. Results strongly support modified random network theory.     

Changes in alkali-silicate glasses induced with electron irradiation

Potassium-silicate, potassium–calcium-silicate and sodium–potassium-silicate glasses were irradiated with low- (1600 eV) and high-energy (50 keV) electrons. Induced changes were recorded by EPMA and XPS. Structural changes connected with transport of alkali ions are discussed.     

Predicting high-codimension critical transitions in dynamical systems using active learning

Complex dynamical systems, from those appearing in physiology and ecology to Earth system modelling, often experience critical transitions in their behaviour due to potentially minute changes in their parameters. While the focus of much recent work, predicting such bifurcations is still notoriously difficult. We propose an active learning approach to the classification of parameter space of dynamical systems for which the codimension of bifurcations is high. Using elementary notions regarding the dynamics, in combination with the nearest-neighbour algorithm and Conley index theory to classify the dynamics at a predefined scale, we are able to predict with high accuracy the boundaries between regions in parameter space that produce critical transitions.     

Resonances in electron-impact integral excitation cross sections of the magnesium atom

The resonance structure of the integral cross sections of excitation of the magnesium atom by low-energy electrons is analyzed in terms of the R-matrix method. The collision strengths are calculated in terms of the close-coupling approximation with consideration of 25 atomic states, including 13 physical target states and 12 pseudostates. The latter simulate the remaining bound and continuum states of the Mg atom that were not included in the close-coupling expansion explicitly. The positions and widths of the resonances found are determined, and the resonances are classified with respect to the mechanism of formation of short-lived states of the negative Mg ion. The results of calculations are compared with available experimental data.     

Comparison between rigorous theory and WKB-analysis of modes in graded-index waveguides

The mode spectrum and the eigenmodes of inhomogeneous optical waveguides are calculated using the WKB-approximation technique. The results are compared with the predictions of rigorous theory as obtained from direct numerical solutions of Maxwell's equation. Three different index profiles of physical interest are considered. It is found that for most modes the WKB-calculation of the effective index is accurate to better than 0.000 15.     

<Emphasis Type="Italic">R</Emphasis>-matrix calculations of the integral electron-impact excitation cross sections of the ground state of the barium atom

The R-matrix approach has been used to calculate the integral cross sections of excitation of the barium atom by low-energy electrons. The collision strengths have been calculated in the 38-state close coupling approximation, including 16 physical states and 22 pseudostates of the target. The target wave functions have been obtained in terms of the multiconfiguration Hartree-Fock method in the LS coupling scheme. The cross sections calculated exhibit a pronounced resonance structure, which contributes significantly to the total excitation cross sections. The results obtained have been compared to available theoretical and experimental data.     

Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution

Steady flow of an incompressible fluid takes place in a porous formation of spatially variable hydraulic conductivityK. The latter is regarded as a lognormal stationary random space function and Y=ln(K/K _G ), whereK _G is the geometric mean ofK, is characterized by its variance ² and correlation scale I. Exact results are known for the effective conductivityK _eff in one- and two-dimensional flows. In contrast, only a first-order term in a perturbation expansion in ² has been derived exactly for the three-dimensional flow. A conjecture has been made in the past onK _eff for any ², but it was not yet proved exactly. This study derived the exact nonlinear correction, i.e. the termO(⁴) ofK _eff, which is found to be the one resulting from the conjecture, strengthening the confidence in it. It is also shown that the self-consistent approximation leads to the exact results for one-dimensional and two-dimensional flows, but underestimates the nonlinear correction ofK _eff for in the three-dimensional case.