Hybrid femtosecond laser microfabrication to achieve true 3D glass/polymer composite biochips with multiscale features and high performance: the concept of ship‐in‐a‐bottle biochip

True three‐dimensionally (3D) integrated biochips are crucial for realizing high performance biochemical analysis and cell engineering, which remain ultimate challenges. In this paper, a new method termed hybrid femtosecond laser microfabrication which consists of successive subtractive (femtosecond laser‐assisted wet etching of glass) and additive (two‐photon polymerization of polymer) 3D microprocessing was proposed for realizing 3D “ship‐in‐a‐bottle” microchip. Such novel microchips were fabricated by integrating various 3D polymer micro/nanostructures into flexible 3D glass microfluidic channels. The high quality of microchips was ensured by quantitatively investigating the experimental processes containing “line‐to‐line” scanning mode, improved annealing temperature (645°C), increased prebaking time (18 h for 1mm‐length channel), optimal laser power (1.9 times larger than that on the surface) and longer developing time (6 times larger). The ship‐in‐a‐bottle biochips show high capabilities to provide simultaneous filtering and mixing with 87% efficiency in a shorter distance and on‐chip synthesis of ZnO microflower particles.     

Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation     

The concept of hyperellipticity on surfaces with nodes

A surface with nodes X is hyperelliptic if there exists an involution such that the genus of X/〈h〉 is 0. We prove that this definition is equivalent, as in the category of surfaces without nodes, to the existence of a degree 2 morphism satisfying an additional condition where the genus of Y is 0. Other question is if the hyperelliptic involution is unique or not. We shall prove that the hyperelliptic involution is unique in the case of stable Riemann surfaces but is not unique in the case of Klein surfaces with nodes. Finally, we shall prove that a complex double of a hyperelliptic Klein surface with nodes could not be hyperelliptic.     

Parametrization of the Teichmüller space of bordered surface NEC groups

A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endowed with dianalytic structure. These surfaces can be seen as a generalization of Riemann surfaces.
Fundamental polygons play an important role in the study of parametrizations of the Teichmuller space of NEC groups.
In this work we construct a class of right-angled polygons which are fundamental regions of bordered surface NEC groups. The free parameters used in the construction of the polygons give a parametrization of the Teichmuller space. From the parameters we obtain explicit matrices of the generators of the groups. Finally, we give examples to exhibit how different relations between the parameters reflect the existence of automorphisms on the quotient surfaces.     

On the numerical solution of the Klein‐Gordon equation

A predictor–corrector (P–C) scheme based on the use of rational approximants of second‐order to the matrix‐exponential term in a three‐time level reccurence relation is applied to the nonlinear Klein‐Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double‐soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Approximating a real number by a rational number with a limited denominator: A geometric approach

We consider the problem of determining the rational number which best approximates the real number a and such that its denominator belongs to an interval [b,b^′]. There is a related geometric problem consisting in finding the integer point lying in the vertical domain D of the form {(x,y)∈R²∣b≤x≤b^′} such that the straight line passing through the origin and through this point best approximates the straight line L of slope a passing through the origin. The computation of this point is interlinked with the computation of both the convex hulls of the integer points located above and below the straight line L respectively and lying in the vertical domain D. In the literature, many general convex hull algorithms exist, as the gift wrapping algorithm for example. However, we focus on two interesting approaches to compute these convex hulls which are especially appropriated in our special configuration. The first one mainly uses number theory and runs in O(log(b^′)) time. The other is in line with computational geometry as the method proposed in 1999 by Balza-Gomez et al. [H. Balza-Gomez, J.-M. Moreau, D. Michelucci, Convex hull of grid points below a line or a convex curve, in: DGCI ’99: Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery, Springer, Marne-la-Vallée, France, 1999, pp. 361-374] which runs in O(log(b^′−b)) time. We propose a new method for the computation of these convex hulls which combines number theory and computational geometry. Our method preserves the optimal time complexity and is the first being output sensitive. Indeed, we compute the convex hulls in time linear in their vertex number. Moreover, the resulting algorithm is very simple and so is suitable for implementation.     

Dimension in a Radiative Stellar Atmosphere

Dimensional scales are examined in an extended 3 + 1 Vaidya atmosphere surrounding a Schwarzschild source. At one scale, the Vaidya null fluid vanishes and the spacetime contains only a single spherical 2-surface. Both of these behaviors can be addressed by including higher dimensions in the spacetime metric.     

Bound states of Klein－Gordon equation for ring-shaped harmonic oscillator scalar and vector potentials

Solving Klein－Gordon equation with equal ring-shaped harmonic oscillator scalar and vector potentials, we obtain the exact normalized bound-state wavefunction and energy equation.     

Explicit and exact travelling plane wave solutions of the （2＋1）—dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions,periodic wave solutions,Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein－Gordon equation.