Lie Algebra and Lie Super Algebra for Integrable Couplings of C-KdV Hierarchy

Based on the constructed Lie algebra and Lie super algebra, the integrable couplings and super-integrable couplings of the C-KdV hierarchy are obtained respectively. Furthermore, its super-Hamiltonian structures are presented by using super-trace identity.     

A New Conserved Quantity Corresponding to Mei Symmetry of Tzenoff Equations for Nonholonomic Systems

A new conserved quantity is investigated by utilizing the definition and discriminant equation of Mei symmetry of Tzénoff equations for nonholonomic systems. In addition, the expression of this conserved quantity, and the determining condition induced new conserved quantity are also presented.     

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach.     

A novel realization of the Virasoro algebra in number state space

The group of diffeomorphisms is crucial in quantum computing. Representing it by vector fields over a d-manifold, d?2, accounting for both projective action and conformal symmetry at the quantum mechanical level, requires the direct-sum decomposition of tensor product for non-compact algebras, viable only for su(1,1). As a step towards the solution, a realization of the (d=1) Virasoro algebra Vir∼Diff₊(S(1)) in the universal envelope of su(1,1) (and h(1)) is presented, which is simple in the discrete positive series irreducible unitary representation of su(1,1).     

Extended Power Law and Hall Anomaly of High-Temperature Superconductors

Starting from the free energy of a moving vortex, we obtain the extended power law form of longitudinal resistivity pxx and the analytical form of Hall resistivity pxy.Based on them, we obtain a scaling relation which agrees well with the experimental data of different kinds of high-temperature superconductors. Furthermore, the theoretical results well fit the scaling results.     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Disturbed Nonholonomic Systems

For a nonholonomic mechanics system with the action of small disturbance, the Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type are studied under general infinitesimal transformations of groups in which the generalized coordinates and time are variable. On the basis of the invariance of disturbed nonholonomic dynamical equations under general infinitesimal transformations, the determining equations, the constrained restriction equations and the additional restriction equations of Lie symmetries of the system are constructed, which only depend on the variables t, qs and q^.s. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for a nonholonomic system with the action of small disturbance is investigated, and the Lie symmetrical adiabatic invariants, the weakly Lie symmetrical adiabatic invariants and the strongly Lie symmetrical adiabatic invariants of generalized Hojman type of disturbed nonholonomic systems are obtained. An example is given to illustrate applications of the results.     

Study on infrared multiphoton excitation of the linear triatomic molecule by the Lie-algebra approach

Infrared multiphoton vibrational excitation of the linear triatomic molecule has been studied using the quadratic anharmonic Lie-algebra model, unitary transformations, and Magnus approximation. An explicit Lie-algebra expression for the vibrational transition probability is obtained by using a Lie-algebra approach. The infrared multiphoton vibrational excitation of the DCN molecule is discussed as an example.     

Initial-value Problems for Extended KdV-Burgers Equations via Generalized Conditional Symmetries

We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.     

A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems： Adiabatic Invariants of Generalized Lutzky Type

For a Lagrangian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariant are presented in general infinitesimal transformation groups. On the basis of the invariance of disturbed Lagrangian systems under general infinitesimal transformations, the determining equations of Lie symmetries of the system are constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariant, i.e. generalized Lutzky adiabatic invariants, of a disturbed Lagrangian system are obtained by investigating the perturbation of Lie symmetries t＇or a Lagrangian system with the action of small disturbance. Finally, an example is given to illustrate the application of the method and results.     

Lie algebras of nonstandard relativity groups

It is shown that representations of Lie algebras of the possible nonstandard spacetime symmetry groups may be derived from the representations of the Poincaré group. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.