A qualitative approach to aberrations of optical elements

A qualitative method to analyze wavefront aberrations is presented.Aberrations of the primary type,expressed in their matrix forms,are used to write the generalized ray transfer matrix of an optical component.The aberrations were treated collectively by examining the pseudospectra of an augmented matrix constructed from the aberration matrices.Results show that aberrations can be distinguished and relative strengths pronounced using this qualitative method.     

Mode characteristics of nano-width rectangle resonator

A new method based on Maxwell's equations,ABCD ray matrices,and total internal reflection is proposed to theoretically analyze the characteristics of eigenmodes confined in nano-width rectangle resonators.Using this method,mode wavelengths and indices of transverse and longitudinal modes are obtained.Another method based on the finite difference time domain technique and Pad′e approximation is used to numerically calculate resonant wavelengths,mode field distributions and quality factors.The results of two methods show that the resonant wavelengths obtained from both methods are very close,and the maximum relative error is less than 2%.The mode indices of transverse and longitudinal modes obtained agree well with mode field distribution patterns calculated by finite difference time domain techniques.     

Padé approximant approach to singular properties of quantum gases: the ideal cases

In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.     

Variable Separation and Exact Solutions to Generalized Nonlinear Diffusion Equations

We study in detail a method to find the generalized nonlinear diffusion equations,which can be solved by means of the variable separation approach.A complete list of canonical forms for such equations,which admit the functional separable solutions,is botained and some exact solutions to the resulting equations are described. A number of methods have been proven to be effective for finding symmetry reductions and constructing exact solutions to nonlinear diffusion equations.     

Solitons for a generalized variable-coefficient nonlinear SchrSdinger equation

In this paper, we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear SchrSdinger equation （NLS） with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions, one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results, some previous one- and two- soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one- and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Double Reduction and Exact Solutions of Zakharov–Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the （1＋2）-dimensional Zakharov-Kuznetsov modified equal width （ZK-MEW） equation with power law nonlinearity are constructed by using Noether＇s approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. FinaJ1y, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.     

A Parametric Dynamic Model of Dark Energy

The double complex symmetric gravitational theory is extended to the parametric symmetric gravitational theory by introducing a parameter β. Hence parametric Friedmann-Robertson-Walker equations are obtained and some characters of dark energy in corresponding spaces are discussed by taking different values of β. In our method some previous results can be included as the special case of our results. It is worth noting that some characters of dark energy can be more intuitively described in our model. By analysis, we can predict that the fate of universe would be a Big Rip in the future, and also find that the state parameters for the two different constraint conditions wФ are consistent with the present cosmological observations.     

Routh method of reduction for Birkhoffian systems in the event space

For a Birkhoffian system in the event space, this paper presents the Routh method of reduction. The parametric equations of the Birkhoffian system in the event space are established, and the definition of cyclic coordinates for the system is given and the corresponding cyclic integral is obtained. Through the cyclic integral, the order of the system can be reduced. The Routh functions for the Birkhoffian system in the event space are constructed, and the Routh method of reduction is successfully generalized to the Birkhoffian system in the event space. The results show that if the system has a cyclic integral, then the parametric equations of the system can be reduced at least by two degrees and the form of the equations holds. An example is given to illustrate the application of the results.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

A unified algorithm for target detection and tracing based on data of array sensors

A unified method for target detection and tracing based on data from sensors of array is presented in order to improve detection and tracking abilities of the weak targets with low signal-to-noise ratio. Assuming that the multiple targets are uncorrelated each other and the number of the targets is known a priori, the status of the targets can be estimated with the maximum a-posteriori （MAP） method directly through the sensors data. The proposed method is different from the classical method, by which it can detect and track targets simultaneously by adding the target＇s signal energy information besides its direction of arrivM（DOA） information. Simulated and sea trial data results show that the detection and tracing capabilities of weak targets can be improved and wrong tracing and missing tracing problems, which exist in the classical tracing method when it is faced with the crossing targets, can be resolved by the proposed method.     

Finite Space Complete Basis Method： Precision Computation of High-Resolutior Spectrum near Ionization Threshold

A new method is proposed to describe quantum dynamical processes in finite space by using of a set ofdiscretized complete bases. In this method, the finite space complete basis is obtained by solving the self-consistent field equation with reflecting boundary conditions. Hence, both negative and positive orbital energies can be obtained. Such method can be used in systems which involve dynamics only in the reaction zone, i.e., in a finite space. To illustrate the validity of the method, we present two examples： theoretical calculation of the high excited states spectra including the continuum of sodium and barium.     

Analytic Debye-Grüneisen equation of state for a generalized Lennard-Jones solids

The approximate method to treat the practical quantum anharmonic solids proposed by Hardy,Lacks and Shukla is reformulated with explicit physical meanings.It is shown that the quantum effect is important at low temperature,it can be treated in the harmonic framework.and the anharmonic effect is important at high temperature and tends to zero at low temperature,it can be treated by using a classical approximation.The alternative formulation is easier for various applications,and is applied to a Debye-Grueneisen solid with the generalized Lennard-Jones intermolecular interaction.The expressions for the Debys temperature and Grueneisen parameter as a function of volume are analytically derived.The analytic equation of state is applied to predict the thermodynamic properties of solid xenon at normal-pressure with the nearest-neighbour Lennard-Jones interaction,and is further applied to research the properties of solid xenon and krypton at high pressure by using an all-neighbour Lennard-Jones interaction.The theoretical results are in agreement with the experiments.     

Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions

<正>The adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions is investigated in this paper.Based on Lyapunov stability theory and Barbalat’s lemma,generalized matrix projective lag synchronization criteria are derived by using the adaptive control method.Furthermore,each network can be undirected or directed,connected or disconnected,and nodes in either network may have identical or different dynamics.The proposed strategy is applicable to almost all kinds of complex networks.In addition,numerical simulation results are presented to illustrate the effectiveness of this method,showing that the synchronization speed is sensitively influenced by the adaptive law strength,the network size,and the network topological structure.     

Integrability classification and exact solutions to generalized variable-coefficient nonlinear evolution equation

This paper is concerned with the generalized variable-coefficient nonlinear evolution equation(vc-NLEE).The complete integrability classification is presented,and the integrable conditions for the generalized variable-coefficient equations are obtained by the Painlev′e analysis.Then,the exact explicit solutions to these vc-NLEEs are investigated by the truncated expansion method,and the Lax pairs(LP) of the vc-NLEEs are constructed in terms of the integrable conditions.     

Applying the reference-wavelength method to improve the precision of glucose measurement by near infrared spectroscopy

The reference-wavelength method is proposed to diminish the influence of noises on glucose measurement by differentially processing two signals at the reference and measuring wavelengths.At the reference wavelength,the radiation intensity is insensitive to the changes of glucose concentration.Therefore,it can be used as the internal reference to estimate the noise and then to extract the effective glucose signal at the other wavelengths.The validation experiments are constructed in the non-scattering samples with the reference wavelength of glucose at 1525 nm.The results show that the reference-wavelength-based glucose-specific signal extracting method can largely improve the glucose prediction precision from 17.56 to 8.87mg/dL in the two-component experiment and from 26.82 to 9.94 mg/dL in the three-component experiment.     

Dynamic behavior of gas bubble in single bubble sonoluminescence—vibrator model

Single bubble sonoluminescence is a process of energy transformation from sound to light. Therefore the motion equations of near spherical vibration of a gas bubble in an incompressible and viscous liquid can be deduced by Lagrangian Equation with dissipation function when the bubble is considered as a vibrator surrounded by liquid. The analytical solutions in the bubble expanding, collapsing and rebounding stages can be obtained by solving these motion equations when some approximations are adopted. And the dynamic behaviors of the bubble in these three stages are discussed.     

Construction of Different Kinds of Atomic and Molecular Orbitals Using Complete Orthonormal Sets of ψ^α-ETO in Single Exponent Approximation

Using complete orthonormal sets of ψ^α-exponential type orbitals in single exponent approximation the new approach has been suggested for construction of different kinds of functions which can be useful in the theory of linear combination of atomic orbitals. These functions can be chosen properly according to the nature of the problems under consideration. This is rather important because the choice of the basis set may be play a crucial role in applications to atomic and molecular problems. As an example of application, different atomic orbitals for the ground states of the neutral and the first ten cationic members of the isoelectronic series of He atom are constructed by the solution of Hartree-Fock-Roothaan equations using ψ^1, ψ^0 and ψ^-1 basis sets. The cMculated results are close to the numerical Hartree-Fock values. The total energy, expansion coefficients, orbital exponents and virial ratio for each atom are presented.     

Breathers and solitons for the coupled nonlinear Schr?dinger system in three-spine α-helical protein

We mainly investigate the variable-coefficient 3-coupled nonlinear Schr?dinger(NLS) system, which describes soliton dynamics in the three-spine α-helical protein with inhomogeneous effect. The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly. The Hirota method is used to solve the constant coefficient NLS equation, and then we get the one-and two-breather solutions of the variable-coefficient NLS equation. The results show that, in the background of plane waves and periodic waves, the breather can be transformed into some forms of combined soliton solutions. The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail. Our results are helpful to study the soliton dynamics inα-helical protein.     

Adomian decomposition method and Padé approximants for solving the Blaszak--Marciniak lattice

The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.     

Relaxation theory of spin-3/2 Ising system near phase transition temperatures

<正>Dynamics of a spin-3/2 Ising system Hamiltonian with bilinear and biquadratic nearest-neighbour exchange interactions is studied by a simple method in which the statistical equilibrium theory is combined with the Onsager's theory of irreversible thermodynamics.First,the equilibrium behaviour of the model in the molecular-field approximation is given briefly in order to obtain the phase transition temperatures,i.e.the first- and second-order and the tricritical points.Then,the Onsager theory is applied to the model and the kinetic or rate equations are obtained.By solving these equations three relaxation times are calculated and their behaviours are examined for temperatures near the phase transition points.Moreover,the z dynamic critical exponent is calculated and compared with the z values obtained for different systems experimentally and theoretically,and they are found to be in good agrement.