On the Recursive Sequence

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation?in the title is investigated, where all the coefficients are nonnegative real numbers. The paper is motivated by an open problem proposed by Ladas [Open problems and conjectures, J. Differ. Equations?Appl., 5 (1999), 211–215].     

A 2 + ɛ approximation algorithm for the k-MST problem

For any ɛ > 0 we give a (2 + ɛ)-approximation algorithm for the problem of finding a minimum tree spanning any k vertices in a graph (k-MST), improving a 3-approximation algorithm by Garg [10]. As in [10] the algorithm extends to a (2 + ɛ)-approximation algorithm for the minimum tour that visits any k vertices, provided the edge costs satisfy the triangle inequality. Research supported by NSF CAREER award NSF CCR-9502747, NSF grants CCR-0205594 and CCR-0098180, an Alfred Sloan Fellowship, and a Packard Fellowship. Research supported by an NSERC Discovery grant.     

The defect character of interface junction lines

Junction lines, where three or more interfaces meet in polycrystalline materials, are analysed from a topological point of view. Using circuit mapping methods, it is shown that, in contiguous polyerystals, the dislocations constituting the interfaces always react at junctions according to topological conservation principles. This conclusion is at variance with recent suggestions in the literature. In addition, it is shown that, in certain circumstances, junction lines can themselves exhibit defect character, i.e., dislocation and/or disclination character. Such defects arise in order to accommodate the coexistence of the abutting crystals. Simple examples are illustrated.     

Structural role and coordination environment of Fe in Fe2O3–PbO–SiO2–Na2O composite glasses

The structural role, coordination geometry and valence of Fe in a series of Fe₂O₃–PbO–SiO₂–Na₂O glasses are studied by means of Fe-K-NEXAFS and EXAFS spectroscopies. Parameters for the study are the concentration of the Fe and Pb-oxides, the SiO₂/Na₂O ratio and the cast temperature. The EXAFS and NEXAFS results reveal that the role of Fe³⁺ depends on the concentration of Fe₂O₃. More specifically, in most of the studied quaternary systems, the Fe³⁺ ion is a glass former, i.e. the Fe atoms belong to FeO₄ tetrahedra that participate in the formation of the glassy network. The role of Fe as an intermediate oxide is identified only in one sample with 20 wt% Fe₂O₃, where ～80 at.% of the Fe atoms are tetrahedrally coordinated with O atoms, while the remaining ～20 at.% of the Fe atoms occupy octahedral sites. It is also revealed that the tetrahedral coordination of Fe in the vitreous matrix is destroyed when a number of parameters is altered, such as the T_cast, the (Fe + Si)/O and the SiO₂/Na₂O ratio.     

Strain accommodation and interfacial structure of AlN interlayers in GaN

The strain accommodation mechanisms at AlN interlayers in GaN, grown by radio‐frequency plasma assisted molecular beam epitaxy, are studied using transmission electron microscopy techniques and atomistic modelling. Interlayers of various thicknesses grown within GaN epilayers deposited on both sapphire and silicon substrates have been employed. Interlayers of thickness below 6 nm do not exhibit line defects although local roughness of the upper interlayer interface is observed as a result of the Al adatom kinetics and higher interfacial energy compared to the lower interface. Above 6 nm, introduction of a ‐type misfit and threading dislocations constitutes the principal relaxation mechanism. Due to strain partitioning between AlN and GaN, threading dislocations adopt inclined zig‐zag lines thus contributing to the relief of alternating compressive‐tensile elastic strain across the AlN/GaN heterostructure. The observed dislocation configurations are consistent with a model of independent motion by climb or ancillary glide in response to their localized three‐dimensional strain environment. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary Value Problems with Compatible Boundary Conditions

If Y is a subset of the space ℝⁿ × ℝⁿ, we call a pair of continuous functions U, V Y-compatible, if they map the space ℝⁿ into itself and satisfy Ux · Vy ≥ 0, for all (x, y) ∈ Y with x · y ≥ 0. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer's fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points P _δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.     

Convergence in a neutral delay differential equation

By using some facts from limiting equations theory we prove that the solution x(.;?), with continuous initial condition ?, of the neutral functional differential equation [x(t)-cx(t-r)]' =-F(x(t))+F(x(t-r)), t>0, where c ε [0,1), r≧0 and F is (not necessarily strictly) increasing. satisfies lim x(t;?) = &;, where &; is the unique root of the algebraic equation [math001]     

A continuous model for non-closed graded systems and its asymptotic character

The Purpose of the paper is to study the behavior and the asymptotic character of a continuous model for non-closed graded systems. Mathematically, the model is described by a system of volterra integral equations. Then by applying known methods based on the theory of the resolvent kernel and limiting equations we analyze the behavior and the limiting structure of system. Finally, we provide necessary and sufficient conditions for the system to be weakly ergodic     

Pyramidal slip in electron beam heated deformed Titanium

The incidence of perfect glide dislocations, moving on parallel pyramidal slip bands on a particular grain boundary of deformed Titanium is studied by means of Transmission Electron Microscopy. Static experiments, performed by using the electron beam as a heating source, proved that slip propagation across the interface is possible when the angle of intersections between the activated slip planes of the incoming and the outgoing dislocations with the boundary plane is minimised. Additionally, the Burgers vector of the residual dislocations left in the boundary after slip transmission occurred should also be minimised. Due to their very small Burgers vector, residual dislocations are visualised with satisfactory results by an image simulation method.     

Stable steady state of some population models

Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=–x+D(x _t ), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.