Application of Eigenkets of Bosonic Creation Operator in Deriving Some New Formulas of Associated Laguerre Polynomials

Using the resolution of unity composed of bosonic creation operator＇s eigenkets and annihilation operator＇s un-normalized eigenket, which is a new quantum mechanical representation in contour integration form, we derive new contour integration expression of associated Laguerre polynomials L^ρm （｜z｜^2） and its generalized generating function formula. A series of recursive relations regarding to L^ρm （｜z｜^2） are also deduced in the context of the Fock representation by algebraic method.

Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials

The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.

A NOTE ON SOBOLEV ORTHOGONALITY FOR LAGUERRE MATRIX POLYNOMIALS

Abstract. Let {L（Ln^（A,λ）（x）}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0,∞） byLn^（A,λ）（x）=n!/（-λ）^n ∑nk-0（-λ）^k/k!（n-k）!（A＋I）n[（A＋I）k]^-1x^k,where A ∈ C^r×r. It is known that {Ln^（A,λ）（x）}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re（z） 〉 -1 for every z E or（a）. In this note we show that forA such that σ（A） does not contain negative integers, the Laguerre matrix polynomials Ln^（A,λ）（x） are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases： the above matrix case and the known scalar case.

Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

CERTAIN ASYMPTOTIC EXPANSIONS FOR LAGUERRE POLYNOMIALS AND CHARLIER POLYNOMIALS

Here proposed are certain asymptotic expansion formulas for Ln(w-1)(λz) and Cn(ω)(λz) in whichO(λ) and n = 0(λ1/2 )(λ→∞) , z being x complex number. Also presented are certain estimates for the remainders(error bounds) of the asymptotic expansions within the regions D1( - ∞<Rez≤1/2 (ω/λ) and D2(1/2 (ω/λ)≤Re.'C00)? respectively.

SOLUTIONS OF A SYSTEM OF FORCED BURGERS EQUATION IN TERMS OF GENERALIZED LAGUERRE POLYNOMIALS

In this article, we obtain explicit solutions of a linear PDE subject to a class of ra-dial square integrable functions with a monotonically increasing weight function｜x｜n-1eβ｜x｜2/2,β ≥ 0, x ∈ Rn. This linear PDE is obtained from a system of forced Burgers equation via the Cole-Hopf transformation. For any spatial dimension n＆gt;1, the solution is expressed in terms of a family of weighted generalized Laguerre polynomials. We also discuss the large time behaviour of the solution of the system of forced Burgers equation.

Pseudo Laguerre Matrix Polynomials, Operational Identities and Quasi-Monomiality

The main purpose of this paper is to introduce the matrix extension of the pseudo Laguerre matrix polynomials and to explore the formal properties of the operational rules and the principle of quasi-monomiality to derive a number of properties for pseudo Laguerre matrix polynomials.

On the Numerical Solution of Singular Integral Equation with Degenerate Kernel Using Laguerre Polynomials

In this paper, we derive a simple and efficient matrix formulation using Laguerre polynomials to solve the singular integral equation with degenerate kernel. This method is based on replacement of the unknown function by truncated series of well known Laguerre expansion of functions. This leads to a system of algebraic equations with Laguerre coefficients. Thus, by solving the matrix equation, the coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed method.

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Improving Video Watermarking through Galois Field GF(2^(4)) Multiplication Tables with Diverse Irreducible Polynomials and Adaptive Techniques

Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity.This study delves into the integration of Galois Field(GF)multiplication tables,especially GF(2^(4)),and their interaction with distinct irreducible polynomials.The primary aim is to enhance watermarking techniques for achieving imperceptibility,robustness,and efficient execution time.The research employs scene selection and adaptive thresholding techniques to streamline the watermarking process.Scene selection is used strategically to embed watermarks in the most vital frames of the video,while adaptive thresholding methods ensure that the watermarking process adheres to imperceptibility criteria,maintaining the video's visual quality.Concurrently,careful consideration is given to execution time,crucial in real-world scenarios,to balance efficiency and efficacy.The Peak Signal-to-Noise Ratio(PSNR)serves as a pivotal metric to gauge the watermark's imperceptibility and video quality.The study explores various irreducible polynomials,navigating the trade-offs between computational efficiency and watermark imperceptibility.In parallel,the study pays careful attention to the execution time,a paramount consideration in real-world scenarios,to strike a balance between efficiency and efficacy.This comprehensive analysis provides valuable insights into the interplay of GF multiplication tables,diverse irreducible polynomials,scene selection,adaptive thresholding,imperceptibility,and execution time.The evaluation of the proposed algorithm's robustness was conducted using PSNR and NC metrics,and it was subjected to assessment under the impact of five distinct attack scenarios.These findings contribute to the development of watermarking strategies that balance imperceptibility,robustness,and processing efficiency,enhancing the field's practicality and effectiveness.

A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.

The Study of Root Subspace Decomposition between Characteristic Polynomials and Minimum Polynomial

Let Abe the linear transformation on the linear space V in the field P, Vλibe the root subspace corresponding to the characteristic polynomial of the eigenvalue λi, and Wλibe the root subspace corresponding to the minimum polynomial of λi. Consider the problem of whether Vλiand Wλiare equal under the condition that the characteristic polynomial of Ahas the same eigenvalue as the minimum polynomial (see Theorem 1, 2). This article uses the method of mutual inclusion to prove that Vλi=Wλi. Compared to previous studies and proofs, the results of this research can be directly cited in related works. For instance, they can be directly cited in Daoji Meng’s book “Introduction to Differential Geometry.”

A coupled Legendre-Laguerre polynomial method with analytical integration for the Rayleigh waves in a quasicrystal layered half-space with an imperfect interface

The Laguerre polynomial method has been successfully used to investigate the dynamic responses of a half-space.However,it fails to obtain the correct stress at the interfaces in a layered half-space,especially when there are significant differences in material properties.Therefore,a coupled Legendre-Laguerre polynomial method with analytical integration is proposed.The Rayleigh waves in a one-dimensional(1D)hexagonal quasicrystal(QC)layered half-space with an imperfect interface are investigated.The correctness is validated by comparison with available results.Its computation efficiency is analyzed.The dispersion curves of the phase velocity,displacement distributions,and stress distributions are illustrated.The effects of the phonon-phason coupling and imperfect interface coefficients on the wave characteristics are investigated.Some novel findings reveal that the proposed method is highly efficient for addressing the Rayleigh waves in a QC layered half-space.It can save over 99%of the computation time.This method can be expanded to investigate waves in various layered half-spaces,including earth-layered media and surface acoustic wave(SAW)devices.

Linear Functional Equations and Twisted Polynomials

A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view a number of algorithms.

多模型块结构Laguerre函数预测控制在再热汽温系统中的应用

为了降低传统预测控制计算负荷并解决被控对象非线性的问题,融合Laguerre函数、块结构思想与多模型控制技术,提出了一种新型预测控制方法,并将其应用于超临界机组再热汽温系统.针对实际再热汽温系统经济性差的问题,通过改进传统预测控制目标函数,提出了集成经济性的性能指标.仿真结果表明,多模型块结构Laguerre函数预测控制方法可以完全关闭喷水阀门以提高机组经济性,在不同负荷点上均可得到优良的控制效果,计算负荷明显降低.工程应用结果表明,无论是在稳定负荷还是变负荷情况下,该方法均可以有效减小再热汽温波动范围,减少事故喷水量,提高再热汽温系统的安全性、稳定性和经济性.

面向航路规划的Laguerre图构造算法

Voronoi图是一种用于无人机航路规划的图形算法,其得到的初始航路为相邻威胁中心连线的垂直平分线,因而会穿越覆盖范围较广的威胁源。引入计算几何学中的Laguerre图用于航路规划,证明了当两个威胁区域不相交时,Laguerre图生成的初始航路必然从它们之间的空隙内穿过。针对Laguerre图生成算法不易实现的问题,提出一种基于Delaunay图的Laguerre图构造算法,其时间复杂度为线性对数阶。仿真结果证明了Laguerre图在解决航路规划问题上的有效性,所提构造算法的运行时间能够满足在线规划的要求。

基于Laguerre函数模型的预测控制算法

在简述基于Ｌａｇｕｅｒｅ函数近似模型的自适应预测控制原理、算法及其特点的基础上，将现有的多步预测、单步优化的控制算法扩展为多步预测、多步优化的控制算法，并与传统的广义预测控制（ＧＰＣ）作了比较研究，结果表明基于Ｌａｇｕｅｒｅ函数模型的预测控制方法对变时延。

基于Laguerre模型的过热汽温自适应预测PI控制系统

针对火电厂锅炉过热汽温控制的特点,设计1种基于Laguerre模型的自适应预测PI控制器。该预测控制器采用对时延具有良好逼近能力的正交Laguerre函数模型作为预测模型,利用带遗忘因子的最小二乘法在线辨识Laguerre预测模型的系数,以提高系统适应工况变化的能力;滚动优化指标采用比例积分型结构,以提高系统的快速性和鲁棒性。通过对具有严重参数不确定性、扰动以及大迟滞的电厂过热汽温被控对象进行仿真研究,结果表明该方法能够很好地适应对象特性的变化,且控制系统的性能比常规串级控制系统有较大提高。

基于Laguerre图的自优化A-Star无人机航路规划算法

为了降低无人机航路规划的运算量,减少规划时间,确保算法对于任意形状威胁区域和地形的适应性以及所规划航路的准确性,提出了一种新颖的LA-Star算法用于无人机航路规划。首先把威胁区域和禁飞区域简化为圆形,利用Laguerre图算法进行航路预规划,在此基础上简化二次规划空间的范围,之后恢复威胁区域和禁飞区域的真实形状,在简化后的规划空间内使用改进A-Star算法实施二次航路规划,最后对生成的航路进行自优化处理。仿真结果证明了LA-Star算法满足航路规划的实时性和准确性要求。