Recursion Formulae for Obtaining Surfaces with Constant Mean Curvature in R2,1

Though the B(a)cklund transformation on time-like surfaces with constant mean curvature surfaces in R2,1 has been obtained,it is not easy to obtain corresponding surfaces because the procedure of solving the related integrable system cannot be avoided when the B(a)cklund transformation is used.For sake of this,in this article,some special work is done to reform the B(a)cklund transformation to a recursion formula,by which we can construct time-like surfaces with constant mean curvature form known ones just by quadrature procedure.     

Collision dynamics of high-speed droplets upon layers of variable thickness

The collision dynamics between a droplet and a film has been studied with high-impact energy that can be grouped in a dimensionless Weber number, We, as normalized by surface energy. To accomplish this, we have developed a technique based on cutting of a high-speed jet, which can generate a single droplet with speed up to 23 m/s and We on the order of thousands. It was found that the boundaries indicating the occurrence of a central jet and that of a secondary droplet disintegrated from the jet decreased monotonically with increased dimensionless film thickness, H, and remained constant when the film thickness was larger than the crater depth. However, the transition designating multiple droplets that are originated from a central jet shows a non-monotonic trend with the variation of H, with a minimum We being at H ≈ 3, which is about the maximum crater depth, owing to a tuning behavior. The critical We for splashing that occurs at an early phase immediately after the impact is relatively sensitive to the film thickness only when H is between 1 and 2, which increases with reduced H. At large We (≳2,570 for high H), the ejected crown is closed to form a bubble and the transition boundary reveals a similar dependence on H as that for creation of a central jet.     

Mixing of cohesive particles in a shear cell

This paper discusses two series of experiments performed in a shear cell device with six different amounts of silicone oils and using 2-mm soda lime beads as the granular materials. The first series of experiments were mixing experiments, and the developments of mixing layer thicknesses were measured. The second series of experiments had the same experimental conditions as the first series but used different combinations of colored particles. In the second series of experiments, the motions of granular materials were recorded by a high-speed camera. Using the image processing technology and particle tracking method, the positions and velocities of particles were measured. The self-diffusion coefficient could be found from the history of the particle displacements.     

In-plane free vibration of a single-crystal silicon ring

In this paper the natural frequencies and the associated mode shapes of in-plane free vibration of a single-crystal silicon ring are analyzed. It is found that the Si(1 1 1) ring is two-dimensionally isotropic in the (1 1 1) plane for elastic constants but three-dimensionally anisotropic, while the Si(1 0 0) ring is fully anisotropic. Hamilton’s principle is used to derive the equations of vibration, which is a set of partial differential equations with coefficients being periodic in polar variable. Expressing the radial and tangential displacements in sinusoidal form with non-predetermined amplitudes, and through the integration with respect to the circumferential variable, the original governing equations in partial differential form can be converted into the amplitude equations in ordinary differential form. The exact expressions for frequencies and mode shapes are obtained. It is found that for Si(1 0 0) rings the frequencies of a pair of modes, which are equal for an isotropic ring, split due to the anisotropic effect only for the second in-plane vibration mode. The phenomena of frequency splitting and degenerate modes can be proved either based on the conservation of averaged mechanical energy or by the concept of crystallographic symmetry groups. When the single-crystal silicon is replaced by the polycrystalline silicon, which is isotropic in elastic constants, the derived equations for frequencies correctly predict the vanishing of the phenomenon of frequency splitting.     

Synthetic Applications of 3‐(Phenylsulfinyl)‐3‐sulfolene

The title compound 2 was readily prepared by MCPBA oxidation of the sulfide 1. Thermal desulfonylation of 2 gave the sulfoxide‐substituted diene 3. The Diels‐Alder reactions of 3 with various dienophiles gave the cyclized products 4‐9 in good yields. The regiochemistry was found to be dominated by the phenylsulfinyl group, but could be reversed by the presence of a Lewis acid. Deprotonation of 2 by BuLi, followed by the reaction with alkyl halides, gave the substituted 2‐sulfolenes 10. A synthetic application of 10 was demonstrated by converting 10e to the bicyclic product 11.     

A NEW DIFFERENCE SCHEME WITH MULTI-TIME LEVELS

In view of making the best use of information coming from past observational data, a new difference scheme with multi-time levels (p>3) is suggested. Some mathematical characteristics of the scheme, which is called the retrospective scheme, are discussed. The numerical results of some examples show that the calculation accuracy of linear and nonlinear advection equations computed with the retrospective scheme is higher than that obtained via the leapfrog scheme. The scheme can be applied to many fields, such as meteorology, engineering physics, astronautics, environment and economy etc, where systematic observations are made normally.     

Self-similar solutions for the anisotropic affine curve shortening problem

Similarity between the roles of the group on the equation for self-similar solutions of the anisotropic affine curve shortening problem and of the conformal group of on the Nirenberg problem for prescribed scalar curvature is explored. Sufficient conditions for the existence of affine self-similar curves are established. Received June 26, 1999 / Accepted January 28, 2000 / Published online December 8, 2000     

Studies on the Chemical Synthesis of Nanosized Nickel Powder and its Stability

The synthesis of nanosized nickel colloids is investigated by using nickel chloride as the precursor, hydrazine as the reducing agent and palladium as the nucleation agent. High purity nickel can be obtained when the reaction temperature is above 80^°C. At a lower reaction temperature, nickel hydroxide is also present in the precipitate form. Starting concentration of nickel ion is a dominant factor in affecting colloidal sizes, while other parameters such as reaction temperature and quantity of poly(vinyl pyrrolidone) (PVP) exhibit only minor effects. Although PVP has some dispersion power, it is not capable of preventing nickel colloids from gradual agglomeration. Our study shows that the subsequent addition of poly(acrylic acid) (PAA) after synthesis can help to stabilize the nickel suspension. This stabilized suspension can then be de-stabilized by changing pH to lower than 4 and vice versa. This reversible stabilization–de-stabilization process is likely due to the behaviour of PVP–PAA complex in the aqueous solutions.     

Collisional source of the second moment and pressure tensor for inhomogeneous rapid granular flows of highly inelastic spheres

By extending the constitutive theories for homogeneous granular flows of highly inelastic spheres by Richman (J. Rheol 33 (1989) 1293), Chou (J. CSME 16-6 (1995) 577), and Chou and Richman (Physica A 259 (1998) 430), the collisional source of the second moment of fluctuation velocity and pressure tensor were developed in this study for inhomogeneous rapid granular flows of identical smooth highly inelastic spheres. The important mean fields in this flow are the solid fraction, mean velocity, and full second moment of fluctuation velocity. The collisional source of second moment and the collisional flux of momentum are based upon an anisotropic Maxwellian velocity distribution function. The constitutive theory was combined with the experimental results measured by Hsiau and Jang (Exp. Thermal Fluid Sci. 17 (1998) 202) so as to determine the profiles of pressure tensor and collision source of second moment in the inhomogeneous rapid granular shear flows of inelastic spheres. The normal pressure discrepancies were also observed.     

Chaotic behaviour of the general symbolic dynamics

This paper extends symbolic dynamics to general cases. Some chaotic properties and applications of the general symbolic dynamics (∑(X),σ) and its special cases are discussed, where X is a separable metric space.