scholarly journals Algebraic treatment of the Pais–Uhlenbeck oscillator and its PT-variant

2020 ◽  
Vol 98 (10) ◽  
pp. 949-952
Author(s):  
Francisco M. Fernández
Keyword(s):  
Differential Equation ◽  
Matrix Representation ◽  
Quadratic Function ◽  
Algebraic Method ◽  
Jordan Form ◽  
Exceptional Points ◽  
Mathematical Properties ◽  
Algebraic Treatment ◽  
Quadratic Hamiltonian

The algebraic method enables one to study the properties of the spectrum of a quadratic Hamiltonian through the mathematical properties of a matrix representation called regular or adjoint. This matrix exhibits exceptional points where it becomes defective and can be written in canonical Jordan form. It is shown that any quadratic function of K coordinates and K momenta leads to a 2K differential equation for those dynamical variables. We illustrate all these features of the algebraic method by means of the Pais–Uhlenbeck oscillator and its PT-variant.

scholarly journals Algebraic treatment of the Bateman Hamiltonian

2021 ◽  
Author(s):  
Francisco M. Fernández
Keyword(s):  
Infinite Number ◽  
Natural Frequencies ◽  
Matrix Representation ◽  
Algebraic Method ◽  
Regular Matrix ◽  
Ladder Operators ◽  
Complex Eigenvalues ◽  
Algebraic Treatment

We apply the algebraic method to the Bateman Hamiltonian and obtain its natural frequencies and ladder operators from the adjoint or regular matrix representation of that operator. Present analysis shows that the eigenfunctions compatible with the complex eigenvalues obtained earlier by other authors are not square integrable. In addition to this, the ladder operators annihilate an infinite number of such eigenfunctions.

scholarly journals On monotonicity and search strategies in face-based copositivity detection algorithms

2021 ◽  
Author(s):  
B. G.-Tóth ◽  
E. M. T. Hendrix ◽  
L. G. Casado
Keyword(s):  
Research Question ◽  
Quadratic Function ◽  
Mixed Integer ◽  
Quadratic Program ◽  
Matrix Methods ◽  
Detection Algorithms ◽  
Standard Simplex ◽  
Mathematical Properties ◽  
The Face ◽  
Spatial Branch And Bound

AbstractOver the last decades, algorithms have been developed for checking copositivity of a matrix. Methods are based on several principles, such as spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the relative interior minima of the standard quadratic program (StQP) and monotonicity. We derive several theoretical properties related to convexity and monotonicity of the standard quadratic function over faces of the standard simplex. We illustrate with numerical instances up to 28 dimensions the use of monotonicity in face-based algorithms. The question is what traversal through the face graph of the standard simplex is more appropriate for which matrix instance; top down or bottom up approaches. This depends on the level of the face graph where the minimum of StQP can be found, which is related to the density of the so-called convexity graph.

scholarly journals An Analytical Study of Weakly Nonlinear Dynamics of a Walters’ Liquid B around a Flexible Sheet Undergoing Super Linear Stretching

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
P. G. Siddheshwar ◽  
A. Chan ◽  
U. S. Mahabaleswar
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equation ◽  
Boundary Layer ◽  
Analytical Study ◽  
Stretching Sheet ◽  
Quadratic Function ◽  
Weakly Nonlinear ◽  
Governing Differential Equation ◽  
Layer Flow ◽  
Analytical Expressions

The paper discusses the boundary layer flow of Walters’ liquid B over a stretching sheet. The stretching is assumed to be a quadratic function of the coordinate along the direction of stretching. The study encompasses within its realm both Walters’ liquid B and second order liquid. The velocity distribution is obtained by solving the nonlinear governing differential equation. Analytical expressions are obtained for stream function and velocity components as functions of the viscoelastic and stretching related parameters. It is shown that the viscoelasticity goes hand in hand with quadratic stretching in enhancing the lifting of the liquid as we go along the sheet.

scholarly journals The bicomplex quantum Coulomb potential problem

2013 ◽  
Vol 91 (12) ◽  
pp. 1093-1100 ◽  
Author(s):  
J. Mathieu ◽  
L. Marchildon ◽  
D. Rochon
Keyword(s):  
Differential Equation ◽  
Coulomb Potential ◽  
Potential Problem ◽  
Orthonormal System ◽  
Mathematical Formulation ◽  
Algebraic Method ◽  
Hamiltonian Operator ◽  
Number System ◽  
Bicomplex Numbers ◽  
Bicomplex Number

Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [Formula: see text], where ξ is a bicomplex number. Following Pauli’s algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrödinger’s time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.

scholarly journals Solution of the Falkner–Skan Equation Using the Chebyshev Series in Matrix Form

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Abdelrady Okasha Elnady ◽  
M. Fayek Abd Rabbo ◽  
Hani M. Negm
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Numerical Method ◽  
Matrix Representation ◽  
Point Of View ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Infinite Domain ◽  
Computational Point

A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.

Exact Solutions for Time-Fractional Ramani and Jimbo—Miwa Equations by Direct Algebraic Method

2020 ◽  
Vol 12 (7) ◽  
pp. 982-988
Author(s):  
Serbay Duran
Keyword(s):  
Differential Equation ◽  
Three Dimensional ◽  
Rational Functions ◽  
Algebraic Method ◽  
Fractional Partial Differential Equations ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Higher Dimensional ◽  
Liouville Derivative ◽  
Three Dimensional Simulations

In this study, the analytical solutions of some nonlinear time-fractional partial differential equations are investigated by the direct algebraic method. The nonlinear fractional partial differential equation (NLfPDE) which is based on the fractional derivative (fd) in the sense of modified Riemann-Liouville derivative is transformed to the nonlinear non-fractional ordinary differential equation. The hyperbolic and rational functions which are contained solutions are obtained for the sixth-order time-fractional Ramani equation and time-fractional Jimbo—Miwa equation (JME) with the help of this technique. In addition, this method can be applied to higher order and higher dimensional NLfPDEs. Finally, three dimensional simulations of some solutions are given.

scholarly journals Hopf Bifurcation Analysis in a Modified Price Differential Equation Model with Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yanhui Zhai ◽  
Ying Xiong ◽  
Xiaona Ma
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Quadratic Function ◽  
Equation Model ◽  
Price Model ◽  
Differential Equation Model ◽  
Normal Form Theory ◽  
Price Differential ◽  
Local Hopf Bifurcation ◽  
Two Delays

The paper investigates the behavior of price differential equation model based on economic theory with two delays. The primary aim of this thesis is to provide a research method to explore the undeveloped areas of the price model with two delays. Firstly, we modify the traditional price model by considering demand function as a downward opening quadratic function, and supply and demand functions both depending on the price of the past and the present. Then the price model with two delays is established. Secondly, by considering the price model with one delay, we get the stable interval. Regarding another delay as a parameter, we studied the linear stability and local Hopf bifurcation. In addition, we pay attention to the direction and stability of the bifurcating periodic solutions which are derived by using the normal form theory and center manifold method. Afterwards, the study turns to simulate the results through numerical analysis, which shows that the provided method is valid.

scholarly journals Special Orthogonal Polynomials in Quantum Mechanics

2019 ◽  
Author(s):  
Abdulaziz D. Alhaidari
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Solution Space ◽  
Type Equation ◽  
Algebraic Method ◽  
Recursion Relations ◽  
New Class ◽  
Continuous And Discrete Spectra ◽  
Discrete Spectra

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.

scholarly journals Logarithmic Quadratic Regression Model for Early Periods of COVID-19 Epidemic Count Data

2020 ◽  
Author(s):  
Daisuke Tominaga
Keyword(s):  
Differential Equation ◽  
Regression Model ◽  
Linear Model ◽  
Generalized Linear Model ◽  
Early Period ◽  
Quadratic Function ◽  
Compartment Model ◽  
Function Model ◽  
Epidemic Curve ◽  
Transition Mechanism

Abstract Background While COVID-19 epidemic has been spreading worldwide, its characteristics are still unclear. The development of good mathematical models for predicting its prevalence and subsiding is strongly expected. The epidemic curve shows how the epidemic increases and subsides. This is the number of persons found infected daily. To express this with a mathematical model, the compartment model such as the SIR model is used generally. However, model parameter values of these ordinary differential equation based models are very sensitive for errors of observed data, and it is often difficult to find a reliable model especially when the amount of data is not sufficient. On the other hand, a regression model with a small number of parameters is more robust against data errors than a highly sensitive nonlinear differential equation model, though, it is not clear what a good regression model is for epidemic data. MethodsWe modeled the initial emerging period of the epidemic curve of COVID-19 in Tokyo with a model that introduces a quadratic polynomial function to the logarithms of the numbers of infected cases, and modeled it with other regression models including the generalized linear model to compare. Results It was shown that the statistical properties of the logarithmic quadratic function model were good even in the early stages of the epidemic, which is generally said to increase exponentially and monotonically. By applying the logarithmic quadratic function model to the data of the number of cases in each country of the world, the starting and the subsiding dates of the epidemic and the total number of cases in each country were estimated. Conclusions Although an epidemic curve in an early period said generally to be exponential, namely linear in the logarithmic space, a quadratic curve regression fits better than the linear and the generalized linear model. These estimates can be informative to reveal the transition mechanism from pre-epidemic to epidemic, and to pandemic.

scholarly journals UNIFIED ALGEBRAIC TREATMENT OF RESONANCE

2005 ◽  
Vol 20 (12) ◽  
pp. 2657-2672 ◽  
Author(s):  
A. D. ALHAIDARI
Keyword(s):  
Bound States ◽  
Algebraic Method ◽  
Complex Scaling ◽  
Complex Angular Momentum ◽  
Momentum Plane ◽  
Angular Momentum Plane ◽  
Rotation Method ◽  
Resonance Energies ◽  
Algebraic Treatment ◽  
Energy Plane

Energy resonance in scattering is usually investigated either directly in the complex energy plane (E-plane) or indirectly in the complex angular momentum plane (ℓ-plane). Another formulation complementing these two approaches was introduced recently. It is an indirect algebraic method that studies resonances in a complex charge plane (Z-plane). This latter approach will be generalized to provide a unified algebraic treatment of resonances in the complex E-, ℓ-, and Z-planes. The complex scaling (rotation) method will be used in the development of this approach. The resolvent operators (Green's functions) are formally defined in these three spaces. Bound states spectrum and resonance energies in the E-plane are mapped onto a discrete set of poles of the respective resolvent operator on the real line of the ℓ- and Z-planes. These poles move along trajectories as the energy is varied. A finite L2 basis is used in the numerical implementation of this approach. Stability of poles and trajectories against variation in all computational parameters is demonstrated. Resonance energies for a given potential are calculated and compared with those obtained by other studies.

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