The Topology of Nontopological Chern–Simons Vortices

The quantized magnetic flux =–4 N(sign ), N=0, ±1, ...of nontopological vortices in nonrelativistic Chern–Simons theory is related to the topological degree of the S² S² mapping defined by lifting the problem to Riemann spheres. Regular solutions with finite degree only arise for rational functions whose topological degree, N, is the common number of zeros and poles on the Riemann sphere, also called their algebraic order.     

Antipodal Theorems for Sets in Rn Bounded by a Finite Number of Spheres

This is a continuation of paper in Adv. Appl. Math. 22 (1999), 219–226, on an antipodal theorem for sets Dⁿ in Rⁿ bounded by a finite number of spheres. Here this theorem is first applied to set-valued mappings from Dⁿ to the boundary of an (n + 1)-cube or a d- dimensional octahedron. Next, the antipodal theorem is reformulated in terms of real continuous functions on Dⁿ, together with applications to the classical theorems of Borsuk–Ulam and Lusternik–Schnirelmann–Borsuk.     

Least-Square Fitting with Spheres

Fitting circles and spheres to given data in is at least relevant in computational metrology (Ref. 1) and reflectrometry (Ref. 2). A new descent algorithm, developed for circles in Ref. 3, is generalized to spheres. Numerical examples are given.     

Rigidity of Minimal Isometric Immersions of Spheres into Spheres

We show two specific uniqueness properties of a certain minimal isometric immersion from S3 into S6. This particular immersion has been extensively studied. We give the first uniqueness results for any such map within the full class of all minimal isometric immersion from S3(1) into S6( ). We also derive an explicit necessary and sufficient condition for linear rigidity of a general minimal isometric immersion from a sphere into a sphere.     

Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of S6

Using a Kaluza–Klein-type procedure, an explicit metric h on an exotic sphere ⁷ is constructed, satisfying the Wiedersehen condition at a set of points diffeomorphic to S ¹. The formulas for the geodesics allows the writing down of formulas for an explicit degree 1 diffeomorphism : S ⁶ S ⁶ that is not isotopic to the identity.     

一类半线性椭圆型方程组的正解

本文利用移动球面法证明了一类半线性椭圆型方程组正解的存在性与不存在性.     

P2O5/SiO2 as an Efficient Reagent for Selective Deprotection of 1,1‐Diacetates under Solvent‐Free Conditions

A mild and efficient method has been developed for the selective deprotection of 1,1‐diacetates of aldehydes in excellent yields by means of the P₂O₅/SiO₂ reagent. Advantages of this method are the use of inexpensive and selective reagent, with high yields in simple operation, and short reaction time under solvent‐free conditions.     

Comparative study of sol-gel-hydrothermal and sol-gel synthesis of titania-silica composite nanoparticles

Titania-silica composite nanoparticles were prepared by sol-gel-hydrothermal and sol-gel routes, respectively, and their physico-chemical and photocatalytic properties were compared. The results of XRD, TEM and BET surface areas showed that sol-gel-hydrothermal route led to anatase titania-silica composite nanoparticles with large specific surface area, but the sol-gel route tended to form mixture of anatase and rutile. The composite nanoparticles prepared by sol-gel-hydrothermal route had better thermal stability against phase transformation from anatase to rutile, agglomeration and particle growth than those prepared by sol-gel route. On the basis of XRD, FT-IR, XPS and ²⁹Si MAS-NMR, a strong interaction was found between SiO₂ and TiO₂, and Ti-O-Si bonds formed during both the two routes. But more Ti-O-Si bonds formed in the composite nanoparticles prepared by sol-gel-hydrothermal route than those prepared by sol-gel route. As a result, the titania-silica composite nanoparticles prepared by sol-gel-hydrothermal route exhibited higher photocatalytic activity in decomposition of methylene blue than that prepared by sol-gel route, and it had excellent photocatalytic activity even after calcined at 1000 °C.     

Study on flow past two spheres in tandem arrangement using a local mesh refinement virtual boundary method

A local mesh refinement virtual boundary method based on a uniform grid is designed to study the transition between the flow patterns of two spheres in tandem arrangement for Re=250. For a small gap (L/D=1.5), the flow field is axisymmetric. As the spacing ratio increases to 2.0, the pressure gradient induces the circumferential fluid motion and a plane‐symmetric flow is constructed through a regular bifurcation. For L/D?2.5, the vortices are periodically shed from the right sphere, but the planar symmetry remains. The case for L/D=3.0 is picked up to give a detail investigation for the unsteady flow. The shedding frequency of vortical structure from the upper side of the right sphere is found to be double of the frequency of the lower side. With the flow spectra of various gaps given, the underlying competitive mechanism between the two shedding frequencies is studied and a critical spacing gap is revealed. Copyright © 2005 John Wiley & Sons, Ltd.