Convergence of a strong variational algorithm for relaxed controls involving a distributed optimal control problem of parabolic type 1

In this paper, we consider a class of Optimal Control problems involving first boundary value problems of parabolic type. A strong variational algorithm has been obtained for solving this class of optimal control problems in a paper by the author and D. W. Reid. It was also shown that any L∞ accumulation points of control sequences generated by the algorithm satisfy a necessary condition for optimality. Since such accumulation points need not exist, it is shown in this paper that control sequences generated by the algorithm always have accumulation points in the sense of control measure, and these accumulation points satisfy a necessary condition for optimality for the corresponding relaxed control problem.     

A linearised singularly perturbed convection–diffusion problem with an interior layer

A linear time dependent singularly perturbed convection–diffusion problem is examined. The convective coefficient contains an interior layer (with a hyperbolic tangent profile), which in turn induces an interior layer in the solution. A numerical method consisting of a monotone finite difference operator and a piecewise-uniform Shishkin mesh is constructed and analysed. Neglecting logarithmic factors, first order parameter uniform convergence is established.     

Pumpless dispensing of a droplet by breaking up a liquid bridge formed by electric induction

Dispensing uniform pico‐to‐nanoliter droplets has become one of essential components in various application fields from high‐throughput bio‐analysis to printing. In this study, a new method is suggested and demonstrated for dispensing a droplet on the top plate with an inverted geometry by using electric field. The process of dispensing droplets consists of two stages: (i) formation of liquid bridge by moving up the charged fluid mass using the electrostatic force between the charges on the fluid mass and the induced charges on the substrate and (ii) its break‐up by the motion of the top plate. Different from conventional electrohydrodynamic methods, electric induction enables the droplets to be dispensed on various surfaces including non‐conducting substrate. The use of capillarity with an inverted geometry removes the need of external pumps or elaborates control for constant flow feed. The droplet diameter has been characterized as a function of the nozzle‐to‐plate distance and the plate moving velocity. The robustness of the present method is shown in terms of nozzle length and applied voltage. Finally, its practical applicability is confirmed by rendering a 19 by 24 array of highly uniform droplets with only 1.8% size variation without use of any active feedback control.     

具有边界摄动的抛物型方程的可解性

本文研究了具有边界摄动的抛物型方程.利用可解性的讨论,得到了原问题的摄动解.     

Existence and uniqueness of weak solutions for a non-uniformly parabolic equation

In this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation.     

Electronic states in a narrow band gap semiconductor microcrystal with parabolic confinement in magnetic field

The energy levels of electrons in a narrow band gap semiconductor microcrystal under the influence of magnetic field are investigated. The confinement potential of microcrystal is approximated as parabolic, and the electron dispersion law is considered within the framework of two-band Kane model. It has been shown that nonparabolicity of dispersion law results in the appearance of the “anharmonic” term in Hamiltonian. The values of magnetic field at which the “anharmonic” term can be considered as perturbation are found. Results of electron energy of nonperturbed Hamiltonian dependencies on values of magnetic field and frequency of microcrystal confinement potential are presented. A comparison of the obtained results with the other cases has been done.     

The zero number diminishing property under general boundary conditions

The so-called zero number diminishing property (or zero number argument) is a powerful tool in qualitative studies of one dimensional parabolic equations, which says that, under the zero- or non-zero-Dirichlet boundary conditions, the number of zeros of the solution $u(x,t)$ of a linear equation is finite, non-increasing and strictly decreasing when there are multiple zeros (cf. Angenent (1988)). In this paper we extend the result to the problems with more general boundary conditions: $u=0$ sometime and $u\ne 0$ at other times on the domain boundaries. Such results can be applied in particular to parabolic equations with Robin and free boundary conditions.     

The exact solution for the free induction decay in a quasi-one-dimensional system in a multi-pulse NMR experiment

The exact solution for the free induction decay in a one-dimensional system in the multi-pulse experiment is obtained at both high and low temperatures in the approximation of nearest neighbor interactions. The experimental investigation is performed on a quasi-one-dimensional system of ¹⁹F nuclear spins in a single crystal of fluorapatite. The theoretical results are in a good agreement with the obtained experimental data.     

A new phase transition in the parabolic Anderson model with partially duplicated potential

We investigate a variant of the parabolic Anderson model, introduced in previous work, in which an i.i.d. potential is partially duplicated in a symmetric way about the origin, with each potential value duplicated independently with a certain probability. In previous work we established a phase transition for this model on the integers in the case of Pareto distributed potential with parameter $\alpha >1$ and fixed duplication probability $p\in (0,1)$: if $\alpha \ge 2$ the model completely localises, whereas if $\alpha \in (1,2)$ the model may localise on two sites. In this paper we prove a new phase transition in the case that $\alpha \ge 2$ is fixed but the duplication probability $p\left(n\right)$ varies with the distance from the origin. We identify a critical scale $p\left(n\right)\to 1$, depending on $\alpha$, below which the model completely localises and above which the model localises on exactly two sites. We further establish the behaviour of the model in the critical regime.