Lie algebraic analysis for the beam transport in the spherical electrostatic analyser

This paper uses the Lie algebraic method to analyse the charged particle trajectories in the spherical electrostatic analyser, and obtains the nonlinear solutions. The results show that the focusing abilities both in the x and y directions of the analyser are almost the same. Moreover, there exist dispersion effects in the x direction, and no dispersion effects in the y direction.

Lie Algebraic Approach to Energy Transfer for Collinear Collision of Two Anharmonic Diatomic Molecules

An anharmonic oscillator algebra model is used to study the collinear collisions of two diatomic molecules. The transition probability for vibration-vibration energy transfer is presented. For an application of the method, we talk about the collision of N2＋CO, N2＋O2, and N2＋N2. Through long time averaging, the transition probability changes to the function of total energy of the system. Comparing the results with the quantum results, we can see that the dynamical Lie algebraic method is useful for describing the anharmonie diatomic molecular collision.

Studies of Rigid Rotor-Rigid Surface Scattering in Dynamical Lie Algebraic Method

The dynamical Lie algebraic method is used for the description of statistical mechanics of rotationally inelastic molecule-surface scattering. It can give the time-evolution operators about the low power of and by solving a set of coupled nonlinear differential equations. For considering the contribution of the high power of and , we use the Magnus formula. Thus, with the time-evolution operators we can get the statistical average values of the measurable quantities in terms of the density operator formalism in statistical mechanics. The method is applied to the scattering of (rigid rotor) by a flat, rigid surface to illustrate its general procedure. The results demonstrate that the method is useful for describing the statistical dynamics of gas-surface scattering.

Solutions of the D-dimensional Schrdinger equation with Killingbeck potential:Lie algebraic approach

Algebraic solutions of the D-dimensional Schrodinger equation with Killingbeck potential are investigated using the Lie algebraic approach within the framework of quasi-exact solvability. The spectrum and wavefunctions of the system are reported and the allowed values of the potential parameters are obtained through the sl（2） algebraization.

Lie algebraic analysis and simulation of high-current pulsed beam transport in a solenoidal lens

Recent calculations of the transport of a high-current beam in a solenoidal lens have made progress to the second order with the Lie algebraic method. A review of the theory and our simulation to realize it will be described. Then we will present the results of simulation. A brief discussion on the space charge effect＇s contribution to the transportation will also be made.

Lie algebraic analysis for the nonlinear transport of intense bunched beam in electrostatic quadrupoles

In this paper, the nonlinear transport of intense bunched beams in electrostatic quadrupoles is analyzed using the Lie algebraic method, and the results are briefly presented of the linear matrix approximation and the second order correction of particle trajectory in the state space. Beam having K-V distribution and Gaussian distribution approximation are respectively considered. A brief discussion is also given of the total effects of the quadrupole and the space charge forces on the evolution of the beam envelope.

Comparison of the PIC model and the Lie algebraic method in the simulation of intense continuous beam transport

Both the PIC (Particle-In-Cell) model and the Lie algebraic method can be used to simulate the transport of intense continuous beams. The PIC model is to calculate the space charge field, which is blended into the external field, and then simulate the trajectories of particles in the total field; the Lie algebraic method is to simulate the intense continuous beam transport with transport matrixes. Two simulation codes based on the two methods are developed respectively, and the simulated results of transport in a set of electrostatic lenses are compared. It is found that the results from the two codes are in agreement with each other, and both approaches have their own merits.

Application of Lie algebraic method to the calculation of rotational spectra for linear triatomic molecules

The Hamiltonian describing rotational spectra of linear triatomic molecules has been derived by using the dynamical Lie algebra of symmetry group U1(4)?U24. After rovibrational interactions being considered, the eigenvalue expression of the Hamiltonian has the form of term value equation commonly used in spectrum analysis. The molecular rotational constants can be obtained by using the expression and fitting it to the observed lines. As an example, the rotational levels ofv 2 band for transition (0200–0110) of molecules N2O and HCN have been fitted and the fitting root-mean-square errors (RMS) are 0.00001 and 0.0014 cm?1, respectively.

LIE ALGEBRAIC STRUCTURE OF AKNS MATRIX SYSTEM

For an AKNS matrix system,Lie algebraic structure and its mastersymmetry are obtained by a purely algebraic approach;and by using the reduced technique,two similar algebraic structures for MKdV and KdV matrix systems are given.

Theoretical study of highly vibrational states of nonlinear triatomic molecules using Lie algebraic approach

The vibrational excitations of bent triatomic molecules are studied by using Lie algebra. The RMS error of fitting 30 spectroscopic data is 1.66 cm-1 for SO2. The results show that the expansion of a molecular algebraic Hamiltonian can well describe the experimental data. And the total vibrational levels can be calculated using this Hamiltonian. At the same time, the potential energy surface can also be obtained with the algebraic Hamiltonian.

Discussion on the Homology Theory of Lie Algebras

Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).

Discussion on the Complex Structure of Nilpotent Lie Groups Gk

Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.

Crossed modules of Lie color algebras

The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the homogeneous components of degree zero of the third cohomology group of Lie color algebras. As an application of this theory, the crossed modules of Witt type Lie color algebras is described, and the result is proved that there is only one equivalent class of the crossed modules of Witt type Lie color algebras when the abelian group Г is equal to Г＋. Finally, for a Witt type Lie color algebra, the classification of its crossed modules is obtained by the isomorphism between the third cohomology group and the crossed modules.

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti- Johnson （G J） hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their ttamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.

Matrix Lie Algebras and Integrable Couplings

Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.

DERIVATIONS AND EXTENSIONS OF LIE COLOR ALGEBRA

In this article,the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer（L）and central extension H^2（L,F）on some Lie color algebras.Meanwhile,they generalize the notion of double extension to quadratic Lie color algebras,a sufficient condition for a quadratic Lie color algebra to be a double extension and further properties are given.

On braided Lie algebras

Let （C, C） be a braided monoidal category. The relationship between the braided Lie algebra and the left Jacobi braided Lie algebra in the category （C, C） is investigated. First, a braided C2-commutative algebra in the category （C, C） is defined and three equations on the braiding in the category （C, C） are proved. Secondly, it is verified that （A, [, ] ） is a left （strict） Jacobi braided Lie algebra if and only if （A, [, ] ） is a braided Lie algebra, where A is an associative algebra in the category （C, C）. Finally, as an application, the structures of braided Lie algebras are given in the category of Yetter-Drinfel＇d modules and the category of Hopf bimodules.

Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F.Zhang and Y.Wang,Phys.Lett.A 360 (2006) 92], three induced Lie algebras are constructed.One of them is obtained by extending Lie bracket,the others are higher- dimensional complex Lie algebras constructed by using linear transformations.The equivalent Lie algebras of the later two with multi-component forms are obtained as well.As their applications,we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.

NON-COMMUTATIVE POISSON ALGEBRA STRUCTURES ON LIE ALGEBRA sl_n(_q) WITH NULLITY M

Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln（Cq^-） are determined.

Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras

Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms,and the generalized analysis method and concise kinematics transfer matrix are obtained.In this study,first,according to the kinematics analysis of serial mechanisms,the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors.Then,based on the standard ideas of Lie operations,the method for kinematics analysis of parallel mechanisms is derived,and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form.Then,according to the mapping relationship between the parallel joints and corresponding equivalent series joints,a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined.A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example.The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.