Lie Algebraic Methods Research Articles

Transport of 3D space charge dominated beams

In this paper we present the theoretical analysis and the computer code design for the intense pulsed beam transport. Intense beam dynamics is a very important issue in low-energy high-current accelerators and beam transport systems. This problem affects beam transmission and beam qualities. Therefore, it attracts the attention of the accelerator physicists worldwide. The analysis and calculation for the intense beam dynamics are very complicated, because the state of particle motion is dominated not only by the applied electromagnetic fields, but also by the beam-induced electromagnetic fields (self-fields). Moreover, the self fields are related to the beam dimensions and particle distributions. So, it is very difficult to get the self-consistent solutions of particle motion analytically. For this reason, we combine the Lie algebraic method and the particle in cell (PIC) scheme together to simulate intense 3D beam transport. With the Lie algebraic method we analyze the particle nonlinear trajectories in the applied electromagnetic fields up to third order approximation, and with the PIC algorithm we calculate the space charge effects to the particle motion. Based on the theoretical analysis, we have developed a computer code, which calculates beam transport systems consisting of electrostatic lenses, electrostatic accelerating columns, solenoid lenses, magnetic and electric quadruples, magnetic sextupoles, octopuses and different kinds of electromagnetic analyzers. The optimization calculations and the graphic display for the calculated results are provided by the code.

Vibrational Spectroscopy of H2O by Lie Algebraic Methods

Vibrational Spectroscopy of H₂O by Lie Algebraic Methods

Generating relations of multi-variable Tricomi functions of two indices using Lie algebra representation

This paper is an attempt to stress the usefulness of the multi-variable special functions. In this paper, we derive certain generating relations involving 2-indices 5-variables 5-parameters Tricomi functions (2I5V5PTF) by using a Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Tricomi and Bessel functions as applications.

Nilpotent class field theory for manifolds

We introduce a nilpotent Hurewicz homomorphism for a topological manifold, following the Lie algebra method in nilpotent class field theory for local and number fields by H. Koch et al. Using Labute-Anick’s results on the determination of the Lie algebra attached to the lower central series of a group, we present explicitly nilpotent class field theory for some two and three dimensional manifolds.

Lie algebra method for solving biological population model

In this paper, the Lie algebraic method is applied to solve biological population models described by time-inhomogeneous birth-death processes. Notwithstanding no obvious symmetry, the solution is expressed by matrix exponentials through suitably generated low-dimensional Lie algebras. This methodology may offer useful insights for other biological and ecological applications.

Infrared-active Vibrational Modes of Fullerene C7O

Using the Lie algebraic method, the stretching fundamental IR active modes of vibrational energies of fullerenes C70 are calculated in the one-dimensional U(2) framework. By constructing the model Hamiltonian with the help of Casimir and Majorana invariant operators in this framework, we calculate the fundamental stretching IR active modes of vibration of fullerene C70 and compare the results with theoretical and experimental values to show the precision of algebraic model.

Equivalence between indirect controllability and complete controllability for quantum systems

We consider a control scheme where a quantum system S is put in contact with an auxiliary quantum system A and the control can affect A only, while S is the system of interest. The system S is then controlled indirectly through the interaction with A. Complete controllability of S+A means that every unitary state transformation for the system S+A can be achieved with this scheme. Indirect controllability means that every unitary transformation on the system S can be achieved. We prove in this paper, under appropriate conditions and definitions, that these two notions are equivalent in finite dimension. We use Lie algebraic methods to prove this result.

Invariants in 3D for Classical Superintegrable Systems in Complex Phase Space

Physical dynamical systems in higher dimensions are always interesting. In this context we present here the possibility of its three-dimensional complex dynamical invariant in extended complex phase space(ECPS). Lie algebraic method is used to study three-dimensional classical superintegrable system on the extended complex phase space. Such complex invariants play an important role in the analysis of complex trajectories, also study of non-hermitian Hamiltonian systems.

A study of vibrational spectra of metallotetraphenyl porphryins: an algebraic approach

Using Lie algebraic method, we report a study of the vibrational frequencies of various metallotetraphenyl porphyrins and its cation radicals, i.e. Cu[TPP], Cu[TPP]+, Ni(TPP), Ni(TPP)+, ClFe(TPP) and ClFe(TPP)+ for 60 vibrational bands each using U(2) algebraic model Hamiltonian. Intermode couplings in algebraic models have been included to achieve improved RMS deviations as well as to get deep insight into detailed spectroscopy for these tetraphenyl porphyrin molecules.

Controllability Aspects of Quantum Dynamics:<newline/>A Unified Approach for Closed and Open Systems

Knowledge about to what extent quantum dynamical systems can be steered by coherent controls is indispensable for future developments in quantum technology. The purpose of this paper is to analyze such controllability aspects for finite dimensional bilinear quantum control systems. We use a unified approach based on Lie-algebraic methods from nonlinear control theory to revisit known and to establish new results for closed and open quantum systems. In particular, we provide a simplified characterization of different notions of controllability for closed quantum systems described by the Liouville-von Neumann equation. We derive new necessary and sufficient conditions for accessibility of open quantum systems modelled by the Lindblad-Kossakowski master equation. To this end, we exploit a well-studied topic of differential geometry, namely the classification of all matrix Lie-groups which act transitively on the Grassmann manifold or the punctured Euclidean space. For the special case of coupled spin-1/2 systems, we obtain a remarkably simple characterization of accessibility. These accessibility results correct and refine previous statements in the quantum control literature.

Spectroscopic Studies on Distorted Structure Nanomolecules by Using Lie Algebraic Model

We have applied Lie algebraic model to distorted structure molecules to determine the vibrational spectra of di erent stretching and bending vibrational modes. The the Lie algebraic model of the Hamiltonian expression is H = E0 + ∑n i=1 AiCi + ∑n i<j AijCij + ∑n i<j λijMij . By using the Lie algebraic method, the stretching vibrational energies of fullerene (C80) are calculated in the one-dimensional [U(2)] framework. Using the model Hamiltonian so constructed, we have calculated the local mode vibrational energy levels of the fullerene (C80) accurately.

Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical, and social applications. In this paper we present the Lie algebraic method, and apply it to three biologically well-motivated examples. The result of this is a solution form that is often highly computationally advantageous.

Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical, and social applications. In this paper we present the Lie algebraic method, and apply it to three biologically well-motivated examples. The result of this is a solution form that is often highly computationally advantageous.

U(2) Lie Algebraic Study of Vibrational Spectra of Fullerene C80 and Its Epoxide C80–O

Using the Lie algebraic method, the stretching vibrational energies of fullerene C80 &amp; its epoxide C80–O are calculated in the one-dimensional U(2) framework. By constructing the model Hamiltonian with the help of Casimir and Majorana invariant operators in this framework, we calculated the local mode vibrational energy levels of fullerene C80 &amp; C80–O.

Entanglement dynamics in a non-Markovian environment: An exactly solvable model

We study the non-Markovian effects on the dynamics of entanglement in an exactly-solvable model that involves two independent oscillators each coupled to its own stochastic noise source. First, we develop Lie algebraic and functional integral methods to find an exact solution to the single-oscillator problem which includes an analytic expression for the density matrix and the complete statistics, i.e., the probability distribution functions for observables. For long bath time-correlations, we see non-monotonic evolution of the uncertainties in observables. Further, we extend this exact solution to the two-particle problem and find the dynamics of entanglement in a subspace. We find the phenomena of `sudden death' and `rebirth' of entanglement. Interestingly, all memory effects enter via the functional form of the energy and hence the time of death and rebirth is controlled by the amount of noisy energy added into each oscillator. If this energy increases above (decreases below) a threshold, we obtain sudden death (rebirth) of entanglement.

Vibrational Frequencies of Buckminsterfullerene: An Algebraic Study

ABSTRACT Using Hamiltonian based on Lie Algebraic method, the normal vibrational modes of Buckminsterfullerene were calculated. The model appeared to describe Carbon-Carbon stretching modes with lower number of algebraic parameters. The dynamical symmetry group of icosahedron molecule was taken into consideration to construct the model Hamiltonian in this framework. Casimir and Majorana invariant operators were also determined accordingly.

Complex invariants in two-dimension for coupled oscillator systems

In order to extract some insight into the features of a dynamical system, we present here the possibility of its complex dynamical invariant. There are many systems which admit complex invariants. To achieve this we use Lie algebraic method to study two dimensional complex systems (coupled oscillator system) on the extended complex phase plane characterized by \(x=x_{1}+ip_{3}, y=x_{2} + ip_{4} , p_x=p_{1} + ix_{3}, p_y=p_{2}+ix_{4}\). Such invariants play an important role in the analysis of complex trajectories with regard to the calculation of semi-classical coherent state propagator in the path integral method.

Design and implementation of SMO for a nonlinear MIMO AHRS

In a low-cost attitude heading reference system (AHRS), the measurements made by MEMS inertial and magnetic sensors are affected by large parameter uncertainties, stochastic noises and unknown disturbances. In this paper, considering the robustness of the sliding mode observers (SMO) against both structured and unstructured uncertainties as well as exogenous inputs, the process of design and implementation of a nonlinear SMO is proposed for a low-cost AHRS. For simultaneous estimation of orientation variables and calibration biases of gyroscopes, a nonlinear and non-affine model of the AHRS is considered. Therefore, based on the Lie-algebraic method, the estimation algorithm is designed for a general class of non-affine nonlinear MIMO systems. In the proposed observer, owing to decreasing the required assumptions for coordinate transformation in recent literatures, the design process of the SMO is simplified. The gain matrices of the proposed SMO are obtained through ensuring the stability and the convergence of estimation errors based on Lyapunov's direct method. The expected tracking performance of the robust state and parameter estimation algorithm compared to that of the extended Kalman filter (EKF) is evaluated through simulations and real experiments of a strapped AHRS on a ground vehicle.

Vibrational spectra of distorted structure Macro &amp;amp; Nano molecules: An algebraic approach

Using the Lie algebraic method the vibrational frequencies of 97 resonances Raman lines (A1g + B1g + A2g + B2g) and 38 infrared bands (Eu) of octaethylporphyrinato-Ni (II) and its mesodeuterated and 15N-substituted derivates and Fullerenes C60 and Cv70 of 7 vibrational bands are calculated using U(2) algebraic Hamiltonian with four fitting algebraic parameters. The results obtained by the algebraic technique have been compared with experimental data; and they show great accuracy.

LEADS-DC: A computer code for intense dc beam nonlinear transport simulation

An intense dc beam nonlinear transport code has been developed. The code is written in Visual FORTRAN 6.6 and has ∼13000 lines. The particle distribution in the transverse cross section is uniform or Gaussian. The space charge forces are calculated by the PIC (particle in cell) scheme, and the effects of the applied fields on the particle motion are calculated with the Lie algebraic method through the third order approximation. Obviously, the solutions to the equations of particle motion are self-consistent. The results obtained from the theoretical analysis have been put in the computer code. Many optical beam elements are contained in the code. So, the code can simulate the intense dc particle motions in the beam transport lines, high voltage dc accelerators and ion implanters.