Incompressible surfaces in handlebodies and isotopies

It is shown that every properly embedded incompressible surface in a handlebody can be constructed by a canonical gluing process. A simple condition is given which asserts that the result of the gluing process is an incompressible surface. A new notion of isotopy is introduced in order to distinguish surfaces belonging to distinct isotopy classes. Several examples (known and new) are constructed.     

The sdfem on Shishkin meshes for linear convection-diffusion problems

Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local error estimates that hold true uniformly in the perturbation parameter , provided only that , where mesh points are used. Numerical experiments support these theoretical results. Received February 19, 1999 / Revised version received January 27, 2000 / Published online August 2, 2000     

Syntheses,Crystal Structures and Thermal Properties of Two Dicadmiun（Ⅱ） Complexes CdeLeX2 （HL = N-（3-Methoxylsalicylidene）-3- dimethylaminopropylamine,X = Cl （1）, NCS （2））

Two Cd（Ⅱ） complexes, Cd2L2Cl2 （1） and Cd2L2（NCS）2 （2） （HL = N-（3-methoxylsalicylidene）-3-dimethylaminopropylamine） were synthesized and determined by EA, IR, TG and single-crystal X-ray diffraction. The crystallographic data are as follows： monoclinic, space group P21/n, a = 9.2710（9）, b = 18.0069（18）, c = 18.5562（19） A^°, β= 99.741（4）°, V = 3053.1（5）, Z = 4,μ = 1.605, F（000） = 1536, R = 0.0264 and wR = 0.0699 for 1; orthorhombic, space group Pca21, a = 16.196（3）, b = 11.506（2）, c = 36.126（7） A^°, V = 6732（2）, Z = 8,μ = 1.428, F（000） = 3264, R = 0.0376 and wR = 0.0877 for 2. There are two geometrically different octahedral Cd（Ⅱ） atoms, with N4O2 and O4Cl2 donor sets for 1 while N4O2 and N2O4 for 2. In the dinuclear Cd（Ⅱ） centers, the Cd（Ⅱ） atoms are held together by two deprotonted phenolate oxygen atoms from tetradentate L^- ligands. The thermal gravity data show two step decompositions with the residues of CdO for two complexes.     

Analytical model for the magnetophoretic capture of magnetic microspheres in microfluidic devices

Magnetic microspheres are used as mobile substrates in micro-total-analysis systems (μTAS), since the particles can be selectively functionalized to attach different bioconjugates and can be precisely manipulated using external magnetic field gradients. A large number of MEMS-based bio-analytical devices employ magnetophoretic separation as an important step during their operation. An analytical technique is proposed in this paper that describes the magnetophoretic transport of magnetic microspheres under an imposed magnetic field when there is a pressure-driven or electroosmotic flow through a microchannel. Successful magnetophoretic capture occurs if the strength of the field-inducing magnetic dipole exceeds a critical value, or if the particles are larger than a critical size. The magnetophoretic separator performance is characterized in terms of capture efficiency. The analysis shows that the capture efficiency is a function of two independent non-dimensional parameters, λ and γ that in turn involve all the physical design and operating parameters of the microfluidic separator, e.g., the dipole strength, particle size and susceptibility, fluid viscosity and velocity, channel height, and the separation of the dipole. Parametric plots of capture efficiency as function of λ and γ helps in choosing the right design and operation parameter of a practical microfluidic separator for a target level of performance.     

Lower bounds for numbers of ABC-hits

By an ABC-hit, we mean a triple (a,b,c) of relatively prime positive integers such that a+b=c and rad(abc)<c. Denote by N(X) the number of ABC-hits (a,b,c) with c?X. In this paper we discuss lower bounds for N(X). In particular we prove that for every ?>0 and X large enough N(X)?exp((logX)^1/2−?).     

Synthesis and Crystal Structure of Mercury(II) Metal Crown Ether with N‐Heterocyclic Carbene Linkage

The mercury(II) metal crown ether ( 2a ) was obtained in high yield by reaction of the carbene precursor 1,2‐bis[N‐(1‐naphthylmethylene)imidazoliumethoxy]benzene dihexafluorophosphate ( 1 ) and Hg(OAc)₂. Addition of NaI to the acetone solution of 2a resulted in precipitation of pale yellow solid 2b . The structures of 2a and 2b were determined by single‐crystal X‐ray diffractometry. Both molecules display a helical conformation with a torsional cycle. The mercury atom in complex 2a is tricoordinated by two intramolecular carbene carbon atoms and an acetate oxygen atom. The mercury atom in complex 2b is tetracoordinated by two intramolecular carbene carbon atoms and two cis‐iodine atoms.     

Photochemical degradation of environmentally persistent perfluorooctanoic acid （PFOA） in the presence of Fe（Ⅲ）

Environmentally persistent and bioaccumulative perfluorooctanic acid （PFOA） was difficult to be decomposed under the irradiation of 254 nm UV light. However, in the presence of 80μmol/L Fe（Ⅲ）, 80% of PFOA with initial concentration of 48μmol/L （20 mg/L） was effectively degraded and 47.8% of fluorine atoms in PFOA molecule were transformed into inorganic fluoride ion after 4 h reaction. Shorter chain perfluorocarboxylic acids bearing C3-C7 and fluoride ion were detected and identified by LC/MS and IC as the degradation products in the aqueous solution. It was proposed that complexes of PFOA with Fe（Ⅲ） initiated degradation of PFOA irradiated with 254 nm UV light.     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.