On some properties of the determinants of tensors

We use the Hilbert?s Nullstellensatz (Hilbert?s Zero Point Theorem) to give a direct proof of the formula for the determinants of the products of tensors. By using this determinant formula and using tensor product to represent the transformations of the slices of tensors, we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices. We also study the determinants of tensors after two types of transposes. We use the permutational similarity of tensors to discuss the relation between weakly reducible tensors and the triangular block tensors, and give a canonical form of the weakly reducible tensors.     

Parameterizing the global attractor of the navier-stokes equations by nodal values

In this paper we give an affirmative answer to the conjecture of Foias and Tern am that the elements of the global attractor of the two dimensional Navier Stokes equations are uniquely determined by their nodal values at a finite number of points in the underlying physical domain. This is kind of a sampling theorem for the elements of the the global attractor of the Navier Stokes equations.     

On finite-time stability for nonlinear impulsive switched systems

This paper is concerned with the finite-time stability problem for switched systems subject to both nonlinear perturbation and impulse effects. The average dwell time approach, combined with the algebraic matrix theory, is utilized to derive a criterion guaranteeing that the state trajectory does not exceed a certain threshold over a pre-specified finite-time interval. The requirement that at least one subsystem should be stable to ensure asymptotic stability is no longer necessary. Moreover, the finite-time stability degree could be positive, which is a relaxed condition for asymptotic stability. A numerical example is presented to illustrate the effectiveness of the proposed method.     

Calculation of the photoelectron spectra under a scaling transform

By solving the time-dependent Schro¨dinger equation, the dependence of photoelectron energy spectra on the binding energy of targets, wavelength and the intensity of laser pulse is exhibited and a scaling law of kinetic energy spectra of both the direct and the rescattered photoelectrons is concluded. The scaling law provides a convenient tool to determine the equivalent photoionization process of various atoms or molecules in various laser fields. The verification of the scaling law by independent methods provides incontestable support to the validity of the scaling law.     

The giant enhancement of Fano-type resonance in a gain-assisted silicon slab array

Giant resonance enhancement is demonstrated to be due to the Fano interference in a grating waveguide composed of gain-assisted silicon slabs. The Fano mode is characterized by its ultra-narrow asymmetric spectrum, different from that of a pure electric or magnetic dipole. The simulation indicates that a sharp Fano-interfered lineshape is responsible for the giant resonance enhancement featuring the small-gain requirements.     

Small amplitude approximation and stabilities for dislocation motion in a superlattice

Starting from the traveling wave solution, in small amplitude approximation, the Sine-Gordon equation can be re- duced to a generalized Duffing equation to describe the dislocation motion in a superlattice, and the phase plane properties of the system phase plane are described in the absence of an applied field. The stabilities are also discussed in the presence of an applied field. It is pointed out that the separatrix orbit describing the dislocation motion as the kink wave may transfer the energy along the dislocation line, keep its form unchanged, and reveal the soliton wave properties of the dislocation motion. It is stressed that the dislocation motion process is the energy transfer and release process, and the system is stable when its energy is minimum.     

First-principles study on the geometric and electronic structures and phase transition of PbZr1-xTixO3 solid solutions

With first-principles virtual-crystal approximation calculations, we systematically investigate the geometric and elec- tronic structures as well as the phase transition of lead zirconate titanate (PbZr1-xTixO3 or PZT) as a function of Ti content for the whole range of 0 ≤ x Ti ≤ 1. It can be found that, with the increase of the Ti content, the PbZr1-xTixO3 solid solu- tions undergo a rhombohedral-to-tetragonal phase transition, which is consistent with the experimental results. In addition, we also show the evolution in geometric and electronic structures of rhombohedral and tetragonal PbZr1-x TixO3 with the increasing content of Ti.     

Formation of hydrophobic coating on PMMA surface using unipolar nanosecond-pulse DBD in atmospheric air

Dielectric barrier discharge (DBD) can modify the material surface and result in complicated physical and chemical reactions to improve the surface hydrophilicity, which is proved to be an effective method for surface modification. Compared with the traditional ac-excitation DBD, the DBD using unipolar pulses can avoid local overheat of microdischarges and can improve discharge efficiency under some conditions. In this paper, DBD excited by repetitive unipolar nanosecond generator was used to improve the hydrophobicity of Plexiglass (PMMA) surface by means of the interaction between air plasma and silicone oil. The output voltage had a rise time of 40 ns and a full width at half maximum of about 70 ns. The surface hydrophobicity of the PMMA, before and after the surface modification, was evaluated via the contact angle measurement under different experimental conditions. The values of the contact angle shown in this paper were the average of eight measured values, and the standard deviations were also calculated. The surface energy including polar and dispersion components was calculated using the measured average contact angles of distilled water and polyethyleneglycol. The results showed that, as the increase of the discharge voltage, the contact angle increased but the surface energy decreased. With the increase of treatment time, the water contact angle of the modified surface increased at the beginning, and it would reach to a maximum at 7.5 min treatment, and then decreased. The effect of pulse frequency on the modification results was different at various treatment times. In addition, the possible physical and chemical reaction among the DBD plasma, silicone oil and the PMMA surface was discussed.     

Thermal noise limit in measuring the gravitational constant G using the angular acceleration method and the dynamic deflection method

A general comparison is made between two methods of measuring the gravitational constant G. The angular acceleration method can avoid the anelasticity effect since the torsion fiber is not twisted. The dynamic deflection method is similar in principle but it does not use feedback, therefore a major noise introduced by the feedback control system in the angular acceleration method can be avoided. Both methods have their advantages and can be performed with the same device. Based on different expressions of G, we have expressed the signal-to-noise ratio and calculated the thermal noise limit for both methods. In order to get a lower thermal noise limit, the dynamic deflection method should avoid resonance.