Lie Algebraic Approach Research Articles

Self-Triggering Based on Lyapunov with Adaptive Control Law for WNCS

This paper addresses the use of Lyapunov functions to self-triggering control implementation. In order to do this a condition, which comes from a sampling technique named Lyapunov sampling, is proposed. The results show the average sampling frequency reduction that can be assessed with this self-triggering method with respect to event and periodic sampling schemes. Using this to dynamically adjust sampling-time, a Lie-algebraic approach is considered to guaranteed closed-loop stability and performance. These results are particularly useful for wireless networked control systems.

LIE SYMMETRY APPROACH TO THE CEV MODEL

Using a Lie algebraic approach we explicitly provide both the probability density function of the constant elasticity of variance (CEV) process and the fundamental solution for the associated pricing equation. In particular we reduce the CEV stochastic differential equation (SDE) to the SDE characterizing the Cox, Ingersoll and Ross (CIR) model, being the latter easier to treat. The fundamental solution for the CEV pricing equation is then obtained following two methods. We first recover a fundamental solution via the {\it invariant solution method}, while in the second approach we exploit Lie classical result on classification of linear partial differential equations (PDEs). In particular we find a map which transforms the pricing equation for the CIR model into an equation of the form $v_{\tau} = v_{yy} - \frac{A}{y^2}v$ whose fundamental solution is known. Then, by inversion, we obtain a fundamental solution for the CEV pricing equation.

Dynamical correlation between quantum entanglement and intramolecular energy in molecular vibrations: An algebraic approach

The dynamical correlation between quantum entanglement and intramolecular energy in realistic molecular vibrations is explored using the Lie algebraic approach. The explicit expression of entanglement measurement can be achieved using algebraic operations. The common and different characteristics of dynamical entanglement in different molecular vibrations are also provided. The dynamical study of quantum entanglement and intramolecular energy in small molecular vibrations can be helpful for controlling the entanglement and further understanding the intramolecular dynamics.

A SIMPLE DERIVATION OF FINITE-TEMPERATURE CFT CORRELATORS FROM THE BTZ BLACK HOLE

We present a simple Lie-algebraic approach to momentum-space two-point functions of two-dimensional conformal field theory at finite temperature dual to the BTZ black hole. Making use of the real-time prescription of AdS/CFT correspondence and ladder equations of the Lie algebra so(2,2) ∼= sl(2,R)<sub>L</sub>⊕sl(2,R)<sub>R</sub>, we show that the finite-temperature two-point functions in momentum space satisfy linear recurrence relations with respect to the left and right momenta. These recurrence relations are exactly solvable and completely determine the momentum-dependence of retarded and advanced two-point functions of finite-temperature conformal field theory.

Su(2) Lie algebra approach for the Feynman propagator of the one-dimensional harmonic oscillator

We evaluate the Feynman propagator for the harmonic oscillator in one dimension. Considering the ladder operators for the Hamiltonian of this system, we construct a set of operators which satisfy the su(2) Lie algebra to obtain Mehler’s formula.

Dissociable morphometric profiles of the affective and cognitive dimensions of alexithymia

Alexithymia (“no words for feelings”) is a psychological construct that can be divided in a cognitive and affective dimension. The cognitive dimension reflects the ability to identify, verbalize and analyze feelings, whereas the affective dimension reflects the degree to which individuals get aroused by emotional stimuli and their ability to fantasize. These two alexithymia dimensions may differentially put individuals at risk to develop psychopathology. However, their neural correlates have rarely been investigated. The aim of the current study was to investigate whether the cognitive and affective alexithymia dimension are associated with unique anatomical profiles. Structural MRI scans of 57 participants (29 males; mean age: 34) were processed using a voxel-based morphometry (VBM) – Diffeomorphic Anatomical Registration Through Exponentiated Lie algebra (DARTEL) approach. Multiple regression analyses were performed to examine the common and specific associations between gray and white matter volume and alexithymia subdimensions. The results revealed that the cognitive dimension was related to lower dorsal anterior cingulate volume. In contrast, the affective alexithymia was associated with lower gray matter volume in the medial orbitofrontal cortex (OFC) and lower white matter volume in the superior longitudinal fasciculus (SLF) near the angular gyrus. No relationship between corpus callosum volume and alexithymia was observed. These results are consistent with the idea that there are two separable neural systems underlying alexithymia. This finding might encourage future research into the link between specific alexithymia subtypes and the development of psychopathology.

ON THE TAKETA BOUND FOR NORMALLY MONOMIAL p-GROUPS OF MAXIMAL CLASS II

We discuss the relationship between the derived length and the number of character degrees in the restricted setting of a normally monomial p-group, G, of maximal class. We continue the Lie algebra approach implemented by Keller, Ragan, and Tims. With a number of technical results, we improve the existing bound, [Formula: see text], to obtain, [Formula: see text].

TRANSITION DENSITY FOR CIR PROCESS BY LIE SYMMETRIES AND APPLICATION TO ZCB PRICING

Using a Lie algebraic approach we explicitly compute the transition density function for the solution of the stochastic differential equation defining the CIR process. Moreover we show how to use such a derivation to recover the well-known formula for the asscoiated ZCB fair price.

Stretching Vibrational IR Active Spectra of Carbon Fullerenes

Molecular vibration spectra are of great significance in the study of molecular structures and characters. Here, an analytical method of Lie algebraic approach is imported into the domain of micro-scaled molecular spectra. The vibrational modes of fullerenes C50, C70 and C80 are calculated considering the C-C bond interactions. The algebraic Hamiltonian describing vibrational spectra of polyatomic molecules has been derived by using the dynamical U(2) Lie algebra. The calculations are performed within the framework of the vibron model with fewer numbers of algebraic parameters. The calculations show a reasonable agreement with previously published calculated and experimental results.

Infrared Spectra of PH3 and NF3: An Algebraic Approach

Molecular spectroscopy is a branch of physics which deals with the interaction of electromagnetic radiation with matter. Using the new theoretical approach, i.e., Lie algebraic approach, we calculate the infrared spectra of phosphine in the range from 3000 cm−1 to 9500 cm−1 and nitrogen trifluoride in the range from 900 cm−1 to 4500 cm−1. The model Hamiltonian constructed seems to describe the P—H and N—F stretching modes accurately with only four numbers of parameters. The present calculation not only predicts the higher overtones but also shows good agreement with the few observed data.

Algebraic approach to quasi-exact solutions of the Dirac-Coulomb problem

The Dirac equation in the presence of Coulomb electrostatic potential is solved and the quasi-exact solutions are obtained via osp(2, 2) algebraization. The Lie-algebraic approach of quasi-exact solvability is applied to the problem and by constructing the matrix representation of the problem, the energy spectrum and thereby the corresponding spinor wave functions are obtained in terms of the polynomial components of osp(2, 2) superalgebra.

SU(1,1) Lie algebraic approach for the evolution of the quantum inflationary universe

Abstract Quantum behavior of scalar fields and vacuum energy density in the inflationary universe are investigated using SU(1,1) Lie algebraic approach. Wave functions describing the evolution of scalar fields that have been thought to have driven cosmic inflation are identified in several possible quantum states at the early stage of the universe, such as the Fock state, the Glauber coherent state, and the SU(1,1) coherent states. In particular, we focus in this research on two important classes of the SU(1,1) coherent states, which are the so-called even and odd coherent states and the Perelomov coherent state. It is shown in the spatially flat universe driven by a single scalar field that the probability densities in all these states have converged to the origin (ϕ = 0, where ϕ is the scalar field) as time goes by. This outcome implies that the vacuum energy density characterized by the scalar field dissipates with time. The probability density in the matter-dominated era converged more rapidly than that in the radiation-dominated era. Hence, we can confirm that the progress of dissipation for the vacuum energy density became faster as the matter era began after the end of the early dominance of radiation. This consequence is, indeed, in agreement with the results of our previous researches in cosmology (for example, see [Chin. Phys. C 35 (2011) 233] and references there in).

A description of Lax type integrable dynamical systems via the Marsden–Weinstein reduction method

A Lie-algebraic approach to constructing nonlinear Lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the Marsden–Weinstein reduction method on canonically symplectic manifolds with group symmetry, is proposed. Its natural relationship with the well known Adler–Kostant–Souriau–Berezin–Kirillov method and the associated R-matrix approach is analyzed.

Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely, the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.

ALGEBRAIC APPROACH TO QUASI-EXACT SOLUTIONS OF THE KLEIN–GORDON–COULOMB PROBLEM

The Klein–Gordon equation in the presence of generalized Coulomb potential is solved and the quasi-exact solutions are obtained via the sl(2) algebraization. The condition of quasi-exact solvability is derived by matching the condition of invariant subspace on the problem. The Lie-algebraic approach of quasi-exact solvability is applied to the problem and the (n+1)×(n+1) matrix for finite values of n is obtained in quite a detailed manner and thereby the finite part of the spectrum is obtained.

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

Abstract We consider a subset S of the complex Lie algebra 𝔰𝔬(n, ℂ) and the cone C(S) of curvature operators which are nonnegative on S. We show that C(S) defines a Ricci flow invariant curvature condition if S is invariant under AdSO(n, ℂ). The analogue for Kähler curvature operators holds as well. Although the proof is very simple and short, it recovers all previously known invariant nonnegativity conditions. As an application we reprove that a compact Kähler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to ℂℙn. Moreover, the methods can also be applied to prove Harnack inequalities.

Lie Algebraic Approach to Nonlinear Integrable Couplings of Evolution Type

Free vibration and flutter analyses of two types of high aspect ratio aircraft wings are presented. The wing is idealised as an assembly of bending-torsion coupled beams using the dynamic stiffness method leading to a nonlinear eigenvalue problem. This problem is solved using the Wattrick-Williams algorithm yielding natural frequencies and mode shapes. The flutter analysis is carried out using the normal mode method in conjunction with generalised coordinates and two-dimensional unsteady aerodynamic theory of Theodorsen. This is essentially a complex eigenvalue problem in terms of both air-speed and frequency. The flutter determinant is solved by an iterative procedure covering a wide range of air-speeds and frequencies. The computed natural frequencies, mode shapes, flutter speeds and flutter frequencies are compared and contrasted for the two type of aircraft wings and some conclusions are drawn.

The Solution of the Spherical Raman–Nath Equation for Free-Electron Laser in the Presence of Ion-Channel Guiding

In this paper, the Wei–Norman Lie algebraic approach has been employed to solve the spherical Raman–Nath equation for free electron laser in the presence of ion channel guiding and helical wiggler exactly. The result can be applied to discuss quantum free electron laser.

Exactly solvable Fokker-Planck equation with time-dependent nonlinear drift and diffusion coefficients – the Lie-algebraic approach

In this paper we have analytically solved the Fokker-Planck equation (FPE) associated with a fairly large class of multiplicative stochastic processes with time-varying nonliner drift and diffusion coefficients, which has wide applicability in various areas of physics, e.g. nonlinear optics and chemical reaction dynamics. By exploiting the dynamical symmetry of the FPE, we apply the Lie-algebraic approach to derive the time-dependent analytical closed-form solutions. The derived solutions fall into two different categories, namely (i) one with a moving absorbing boundary, and (ii) one with a fixed absorbing boundary at the origin, depending upon the model parameters. The corresponding escape (or survival) probabilities are also evaluated analytically. We believe that not only our analytically exact results can serve as standard models upon which the discussion of more complicated problems can be based, but they can also be useful as a benchmark to test approximate numerical or analytical procedures.

Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant

We consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be anisotropic and the magnetic field is applied along z-axis. A canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators. Two classes of real and complex quadratic invariants (constants of motion) are obtained. We employ the Lie algebraic technique to find the most general solution for the wave-function for both real and complex invariants. The quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave-functions in the coherent states and number states are obtained. Some discussion related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself is also given.