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.     

关于调和级数的部分和的推导及证明

中世纪后期,数学家Oresme证明了所有调和级数都是发散的,但是调和级数的拉马努金和存在,且为Euler常数.Euler在1734年利用Newton的成果,首先给出了调和级数的部分和的表达式.通过分析Ross,S.M.对经典概率论问题"优惠券收集问题"的解决方法,得到了调和级数的部分和的不同表达式,并运用数学归纳法,变量代换证明了表达式的正确性.     

High-order splitting methods for separable non-autonomous parabolic equations

We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time-dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods on several numerical examples and present some new improved schemes.