Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.     

Singular finite elements for the sudden-expansion and the die-swell problems

The singular finite element method is used to solve the sudden-expansion and the die-swell problems in order to improve the accuracy of the solution in the vicinity of the singularity and to speed up the convergence. The method requires minor modifications to standard finite element schemes, and even coarse meshes give more accurate results than refined ordinary finite element meshes. Improved normal stress results for the sudden-expansion problem have been obtained for various Reynolds numbers up to 100 using the singular elements constructed for the creeping flow problem. In addition, the normal stresses at the walls appear to be insensitive to the singularity powers used in the construction of the singular basis functions. The die-swell problem is solved using the singular elements constructed for the stick–slip problem. The singular elements accelerate the convergence of the free surface dramatically.     

The Method of Fundamental Solutions for Stokes flows with a free surface

We investigate the use of the Method of Fundamental Solutions (MFS) for solving Stokes flow problems with a free surface. We apply the method to the creeping planar Newtonian extrudate-swell problem and study the effect of the surface tension on the free surface. The results are in good agreement with existing finite element and boundary element solutions. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 667–678, 1998     

Periodicity of equilibrium structures in a time dependent markov model under stochastic environment

The problem of periodicity for a non–homogeneous Markov model in a stochastic environment is studied. The stochastic concept is established through the notion of optional scenarios applied on the transition process. It is proved that the sequence of so–called aggregate structures follows a certain periodic pattern that can split into converging subsequences according to alternative policies. These limits are highly influenced by the different scenarios utilized in the model, but always lie on a convex region that also depends on the pool of alternatives.     

Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method

A singular function boundary integral method (SFBIM) is proposed for solving biharmonic problems with boundary singularities. The method is applied to the Newtonian stick–slip flow problem. The streamfunction is approximated by the leading terms of the local asymptotic solution expansion which are also used to weight the governing biharmonic equation in the Galerkin sense. By means of the divergence theorem the discretized equations are reduced to boundary integrals. The Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers, the values of which are calculated together with the singular coefficients. The method converges very fast with the number of singular functions and the number of Lagrange multipliers, and accurate estimates of the leading singular coefficients are obtained. Comparisons with the analytical solution and results obtained with other numerical methods are also made. Copyright © 2005 John Wiley & Sons, Ltd.     

A far-red emitting probe for unambiguous detection of mobile zinc in acidic vesicles and deep tissue

Imaging mobile zinc in acidic environments remains challenging because most small-molecule optical probes display pH-dependent fluorescence. Here we report a reaction-based sensor that detects mobile zinc unambiguously at low pH. The sensor responds reversibly and with a large dynamic range to exogenously applied Zn²⁺ in lysosomes of HeLa cells, endogenous Zn²⁺ in insulin granules of MIN6 cells, and zinc-rich mossy fiber boutons in hippocampal tissue from mice. This long-wavelength probe is compatible with the green-fluorescent protein, enabling multicolor imaging, and facilitates visualization of mossy fiber boutons at depths of >100 μm, as demonstrated by studies in live tissue employing two-photon microscopy.     

Heat transfer due to harmonic variation in the free stream temperatures in a flat plate exposed on both sides

The heat transfer through an infinite flat plate is studied when the temperatures of the two free streams surrounding it are varying harmonically with time and out of phase, with a delay period τ_d. The configuration is a simplified model for the heat transfer through the separating wall in the isochoric counter-current heat exchanger. The results show that apart from the τ_d effect, the perturbation parameters depend mainly on the cavity passing frequency f. At the thick plate solution, the combined passing frequency–delay time influences are significant only when the dimensionless frequency is smaller than 10. Within this range τ_d affects seriously not only the temperature perturbation amplitudes (which determine the thermal stresses) but also the heat fluxes and the accumulated energy ones. When f ≥ 10, the plate behaves as two separate semi-infinite slabs. Heat penetration delays greater than one cavity passing period may be possible.